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An elementary course of infinitesimal calculus . e diameter is equal to the constant value of 01.Hence the motion is equivalent to the rolling of a circle on theinside of a fixed circle of twice its size. This kind of motion hasbeen considered in Art. 138, Ex. 2, and it has been shewn thatany point P fixed relatively to AB will describe an ellipse, whichin certain cases, viz. when P is on the circumference of the rollingcircle, degenerates into a straight line. Ex. 2. In the linkage known as the crossed parallelogram(see Art. 146, 2°), if the bar AD be held fixed, the instantaneouscentre / for Stock Photo
RM2AXJHJP–An elementary course of infinitesimal calculus . e diameter is equal to the constant value of 01.Hence the motion is equivalent to the rolling of a circle on theinside of a fixed circle of twice its size. This kind of motion hasbeen considered in Art. 138, Ex. 2, and it has been shewn thatany point P fixed relatively to AB will describe an ellipse, whichin certain cases, viz. when P is on the circumference of the rollingcircle, degenerates into a straight line. Ex. 2. In the linkage known as the crossed parallelogram(see Art. 146, 2°), if the bar AD be held fixed, the instantaneouscentre / for
. Fossil plants : for students of botany and geology . Paleobotany. VIl] PERONOSPORITES. 217 in the act of escaping from a lateral pore. This interpretation strikes one as lacking in scientific caution. The sporangia of Hyphochytrium infestans^, as figured by Fischer in Eabenhorst's work bear a close resemblance to those of the fossil. It would seem very probable that Renault's species may be reasonably referred to the Chytridineae, as he proposes.. Fig. 43. 1. Oochytrium Lepldodendri, Een. (After Eenault.) 2. Polyporus vaporarius Fr. yar. succinea. (After Conwentz.) 3. Gladosporites bipar- ti Stock Photo
RMPG16X4–. Fossil plants : for students of botany and geology . Paleobotany. VIl] PERONOSPORITES. 217 in the act of escaping from a lateral pore. This interpretation strikes one as lacking in scientific caution. The sporangia of Hyphochytrium infestans^, as figured by Fischer in Eabenhorst's work bear a close resemblance to those of the fossil. It would seem very probable that Renault's species may be reasonably referred to the Chytridineae, as he proposes.. Fig. 43. 1. Oochytrium Lepldodendri, Een. (After Eenault.) 2. Polyporus vaporarius Fr. yar. succinea. (After Conwentz.) 3. Gladosporites bipar- ti
An elementary course of infinitesimal calculus . , the relative motion ofGD wiU be represented by the rolling of a hyperbola with G, Das foci on an equal hyperbola with A, B as foci. 168. Double Greneration of EpicycUcs as Rou-lettes. As a further example we return to the mechanicalmethod of compounding uniform circular motions, by meansof a jointed parallelogram OQPQ!, referred to in Art. 139. We will suppose for definiteness that the angular velo-cities n, n, of the bars OQ, OQ, have the same sign. The instantaneous centre (7) of the bar QP will be apoiat in QO such that n.QI=n.OQ (1). For t Stock Photo
RM2AXJH8T–An elementary course of infinitesimal calculus . , the relative motion ofGD wiU be represented by the rolling of a hyperbola with G, Das foci on an equal hyperbola with A, B as foci. 168. Double Greneration of EpicycUcs as Rou-lettes. As a further example we return to the mechanicalmethod of compounding uniform circular motions, by meansof a jointed parallelogram OQPQ!, referred to in Art. 139. We will suppose for definiteness that the angular velo-cities n, n, of the bars OQ, OQ, have the same sign. The instantaneous centre (7) of the bar QP will be apoiat in QO such that n.QI=n.OQ (1). For t
. Fossil plants : for students of botany and geology . Paleobotany. 18 PSILOTALES [CH. Sw. (fig. 118) and P. complanatum Sw., flourishes in moist tropical regions of both hemispheres, growing either on soil rich in organic substances or as an epiphyte. Both genera are considered to be more or less saprophytic.. Fig. 118. Psilotum triquetrum (J natural size). A. Synangium. B. Sporophyll after removal of the synangium. (M.S.) Psilotum. The common tropical species P. triquetrum (fig. 118) is characterised by an underground rhizome which forms a confused mass of dark brown branches covered with fi Stock Photo
RMPG16M2–. Fossil plants : for students of botany and geology . Paleobotany. 18 PSILOTALES [CH. Sw. (fig. 118) and P. complanatum Sw., flourishes in moist tropical regions of both hemispheres, growing either on soil rich in organic substances or as an epiphyte. Both genera are considered to be more or less saprophytic.. Fig. 118. Psilotum triquetrum (J natural size). A. Synangium. B. Sporophyll after removal of the synangium. (M.S.) Psilotum. The common tropical species P. triquetrum (fig. 118) is characterised by an underground rhizome which forms a confused mass of dark brown branches covered with fi
An elementary course of infinitesimal calculus . Fig. 105. If the constant of inversion be changed we get a similar curve,which will still be a circle through the centre of similitude 0. 380 INFINITESIMAL CALCULUS. [CH. IX Ex. 2. More generally, the inverse of any circle is a circle.. Fig. 106. Let 0 be the centre of inversion, C the centre of the givencircle, a its radius; and let le = OG--a (6). If, then, we draw any chord OPP through 0, it is known fromGeometry that OP. OP= 00-a?^k (7) Hence P traces out the inverse of the locus of P; i.e. the circleinverts into itself. And by changing the Stock Photo
RM2AXJW0W–An elementary course of infinitesimal calculus . Fig. 105. If the constant of inversion be changed we get a similar curve,which will still be a circle through the centre of similitude 0. 380 INFINITESIMAL CALCULUS. [CH. IX Ex. 2. More generally, the inverse of any circle is a circle.. Fig. 106. Let 0 be the centre of inversion, C the centre of the givencircle, a its radius; and let le = OG--a (6). If, then, we draw any chord OPP through 0, it is known fromGeometry that OP. OP= 00-a?^k (7) Hence P traces out the inverse of the locus of P; i.e. the circleinverts into itself. And by changing the
. Fossil plants : for students of botany and geology . Paleobotany. 392 BENNBTTITALES [CH. certain recent Flowering plants; it is one of the distinguishing features between Gymdeoidea and stems bearing longer fertile shoots with flowers of the Williamsonia type.. Fio. 521. Cycadeoidea (Bennettites) Gibsoniana. A, C, longitudinal sections; a, absciss-layer; b, bracts; pr, ' pericarp'; s, scales, sterile and fertile. B, transverse section of female flower; i, interseminal scales; m, megasporo- phyUs. D, transverse section of seed showing the testa, t, separated by a space, sp, from the embryo. E Stock Photo
RMPG06RW–. Fossil plants : for students of botany and geology . Paleobotany. 392 BENNBTTITALES [CH. certain recent Flowering plants; it is one of the distinguishing features between Gymdeoidea and stems bearing longer fertile shoots with flowers of the Williamsonia type.. Fio. 521. Cycadeoidea (Bennettites) Gibsoniana. A, C, longitudinal sections; a, absciss-layer; b, bracts; pr, ' pericarp'; s, scales, sterile and fertile. B, transverse section of female flower; i, interseminal scales; m, megasporo- phyUs. D, transverse section of seed showing the testa, t, separated by a space, sp, from the embryo. E
An elementary course of infinitesimal calculus . Fig. 151. 197. Derivation of the Iiogarithmic Series, andof Gregorys Series. The following are important applications of the theoremjust proved: 1°. If I a; I < 1, we have1 1+0! = l—x + x — a^ + . ?in 197] INFINITE SERIES. 553 Hence, integrating between the limits 0 and as, log (1+ «) = «-la!+ ia!-ia+ (2), which is the logarithmic series*. The proof applies only for | a; | < 1; and we cannot assertwithout examination that the result is valid for a; = + 1. For x=, the terms on the right-hand of (2) arealternately positive and negative, and Stock Photo
RM2AXJE4Y–An elementary course of infinitesimal calculus . Fig. 151. 197. Derivation of the Iiogarithmic Series, andof Gregorys Series. The following are important applications of the theoremjust proved: 1°. If I a; I < 1, we have1 1+0! = l—x + x — a^ + . ?in 197] INFINITE SERIES. 553 Hence, integrating between the limits 0 and as, log (1+ «) = «-la!+ ia!-ia+ (2), which is the logarithmic series*. The proof applies only for | a; | < 1; and we cannot assertwithout examination that the result is valid for a; = + 1. For x=, the terms on the right-hand of (2) arealternately positive and negative, and
. Fossil plants : for students of botany and geology . Paleobotany. 266 COKDAITEAB [CH. between seeds and vegetative organs, and while it is possible to state with confidence that many of the seeds represented by impressions described as species of Samaropsis and Cordaicarpus are midoubtedly Cordaitean, it is certain that not all seeds refer- able to these genera were borne by Cordaitalean plants. Cordaitean seeds are characterised by certain morphological features recaUiag those found in recent Cycads and in the seeds of Ginkgo as illustrated by species of Cardioca/rpus and some allied types. Stock Photo
RMPG0D1E–. Fossil plants : for students of botany and geology . Paleobotany. 266 COKDAITEAB [CH. between seeds and vegetative organs, and while it is possible to state with confidence that many of the seeds represented by impressions described as species of Samaropsis and Cordaicarpus are midoubtedly Cordaitean, it is certain that not all seeds refer- able to these genera were borne by Cordaitalean plants. Cordaitean seeds are characterised by certain morphological features recaUiag those found in recent Cycads and in the seeds of Ginkgo as illustrated by species of Cardioca/rpus and some allied types.
An elementary course of infinitesimal calculus . a (6). If, then, we draw any chord OPP through 0, it is known fromGeometry that OP. OP= 00-a?^k (7) Hence P traces out the inverse of the locus of P; i.e. the circleinverts into itself. And by changing the constant of inversion we get a similarcurve, and therefore a circle. If, as in the right-hand figure, 0 be within the given circle,the constant in (7) is negative. This means that P and P arenow on opposite sides of 0. 146. Mechanical Inversion. There are various devices by which the inverse of a givencurve can be traced mechanically. 1°. Peau Stock Photo
RM2AXJTM5–An elementary course of infinitesimal calculus . a (6). If, then, we draw any chord OPP through 0, it is known fromGeometry that OP. OP= 00-a?^k (7) Hence P traces out the inverse of the locus of P; i.e. the circleinverts into itself. And by changing the constant of inversion we get a similarcurve, and therefore a circle. If, as in the right-hand figure, 0 be within the given circle,the constant in (7) is negative. This means that P and P arenow on opposite sides of 0. 146. Mechanical Inversion. There are various devices by which the inverse of a givencurve can be traced mechanically. 1°. Peau
. Fungoid and insect pests of the farm. Agricultural pests. XI] Flies 129. Fig. 38. Oat plants attacked by the frit fly (Oecinus frit).. Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original work.. Petherbridge, Frederick Robert. Cambridge : University Press Stock Photo
RMPG0RB4–. Fungoid and insect pests of the farm. Agricultural pests. XI] Flies 129. Fig. 38. Oat plants attacked by the frit fly (Oecinus frit).. Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original work.. Petherbridge, Frederick Robert. Cambridge : University Press
An elementary course of infinitesimal calculus . Fig. 107. Hence if P (or Q) be made to describe any curve, Q (or P)will describe the inverse curve with respect to 0. In particular if, by a link, P be pivoted to a fixed point 8,such that SO = SF, the locus of P is a circle through 0, andconsequently the locus of Q will be a straight line perpendicularto OS. This gives an exact solution of the important mechanicalproblem of converting circular into rectilinear motion by meansof link-work. 2°. Harts Linkage. This consists of a crossed parallelogram A BCD formed H--. Fig. 108. 382 INFINITESIMAL C Stock Photo
RM2AXJT7Y–An elementary course of infinitesimal calculus . Fig. 107. Hence if P (or Q) be made to describe any curve, Q (or P)will describe the inverse curve with respect to 0. In particular if, by a link, P be pivoted to a fixed point 8,such that SO = SF, the locus of P is a circle through 0, andconsequently the locus of Q will be a straight line perpendicularto OS. This gives an exact solution of the important mechanicalproblem of converting circular into rectilinear motion by meansof link-work. 2°. Harts Linkage. This consists of a crossed parallelogram A BCD formed H--. Fig. 108. 382 INFINITESIMAL C
. Fossil plants : for students of botany and geology . Paleobotany. Fig. 8. Piece of coniferous wood in flint, from the Chalk, Croydon. Drawn from a specimen presented to the British Museum by Mr Murton Holmes. In the side view, shown above in the figure, the position of the wood is shown by the lighter portion, with holes, 6, 6, bored by Teredos or some other wood-eating animal. In the end view, below, the wood is seen as an irregular cylinder w, w, embedded in a matrix of flint. Nat. size. The specimen represented in fig. 9 illustrates the almost complete destruction of a piece of wood by Stock Photo
RMPG174C–. Fossil plants : for students of botany and geology . Paleobotany. Fig. 8. Piece of coniferous wood in flint, from the Chalk, Croydon. Drawn from a specimen presented to the British Museum by Mr Murton Holmes. In the side view, shown above in the figure, the position of the wood is shown by the lighter portion, with holes, 6, 6, bored by Teredos or some other wood-eating animal. In the end view, below, the wood is seen as an irregular cylinder w, w, embedded in a matrix of flint. Nat. size. The specimen represented in fig. 9 illustrates the almost complete destruction of a piece of wood by
An elementary course of infinitesimal calculus . which it encloses. The formulae (5) of Art. 137 give, for b = 2a, x= — a cos & — 2a cos 6, y= — asm6- 2a sin ^6, or x-a = -2a{+cosff)cos6, y = -2a{+cos.^6)sva.^0 (8). If we put & = 16 + IT, it appears that tlie pericycloid, referred to the point (a, 0) aspole, has the equation r = 2.a{-oos6) (9), and is therefore a cardioid. The connection between this result and that of Ex. 1, above,will appear in Art. 168. Ex. 4. The four-cusped hypocycloid. If in Art. 137 (6) we put h = a, we get x = ^acoa6 + acosW = acos^0, y = lasm6-as.M = asm^6] Stock Photo
RM2AXK0FT–An elementary course of infinitesimal calculus . which it encloses. The formulae (5) of Art. 137 give, for b = 2a, x= — a cos & — 2a cos 6, y= — asm6- 2a sin ^6, or x-a = -2a{+cosff)cos6, y = -2a{+cos.^6)sva.^0 (8). If we put & = 16 + IT, it appears that tlie pericycloid, referred to the point (a, 0) aspole, has the equation r = 2.a{-oos6) (9), and is therefore a cardioid. The connection between this result and that of Ex. 1, above,will appear in Art. 168. Ex. 4. The four-cusped hypocycloid. If in Art. 137 (6) we put h = a, we get x = ^acoa6 + acosW = acos^0, y = lasm6-as.M = asm^6]
. Fossil plants : for students of botany and geology . Paleobotany. XLV] voltzia; swedenborgia 295 phylls. The species Voltzia recubariensis (Mass.) represented by several vegetative shoots, imperfect cones, cone-scales, and seeds in the Muschelkalk beds of Recoaro^ illustrates the impossibility in the case of sterile specimens of drawing any satisfactory line between Voltzia and Pagiophyllum. Heer instituted the genus Leptostrobus for strobili from Jurassic strata in Siberia agreeing closely in habit and in the form of the megasporophylls with those of VoUzia. The strobili, referred by Heer t Stock Photo
RMPG161E–. Fossil plants : for students of botany and geology . Paleobotany. XLV] voltzia; swedenborgia 295 phylls. The species Voltzia recubariensis (Mass.) represented by several vegetative shoots, imperfect cones, cone-scales, and seeds in the Muschelkalk beds of Recoaro^ illustrates the impossibility in the case of sterile specimens of drawing any satisfactory line between Voltzia and Pagiophyllum. Heer instituted the genus Leptostrobus for strobili from Jurassic strata in Siberia agreeing closely in habit and in the form of the megasporophylls with those of VoUzia. The strobili, referred by Heer t
An elementary course of infinitesimal calculus . Fig. 76. Fig, 77. E<c.i. a. See Fig. 77.The curve is known as the witch of Agnesi. Mx. 5. y2 = a;2|±| (15). There is a node at the origin, and the curve cuts the axis of xagain at (-a, 0). For x>b, and x<-a, y is imaginary. Theline a; = 6 is an asymptote. See Fig. 78, ^^?^- y=^^ (16). This is obtained by putting a = 0 in (15). The loop now shrinksinto a cusp; see Fig. 79. The curve is known as the cissoid. 1. 22 338 INFINITESIMAL CALCULUS. [CH. IX. Stock Photo
RM2AXK7KR–An elementary course of infinitesimal calculus . Fig. 76. Fig, 77. E<c.i. a. See Fig. 77.The curve is known as the witch of Agnesi. Mx. 5. y2 = a;2|±| (15). There is a node at the origin, and the curve cuts the axis of xagain at (-a, 0). For x>b, and x<-a, y is imaginary. Theline a; = 6 is an asymptote. See Fig. 78, ^^?^- y=^^ (16). This is obtained by putting a = 0 in (15). The loop now shrinksinto a cusp; see Fig. 79. The curve is known as the cissoid. 1. 22 338 INFINITESIMAL CALCULUS. [CH. IX.
. Fossil plants : for students of botany and geology . Paleobotany. 246 PTERIDOPHYTA. [CH. main characters of the recent genus will enable the student to appreciate the points of difference and agreement between the extinct and present representatives of the Equisetales.. Fig. 52. Equisetum maximum Lam. A. Fertile shoot with strobilus and sterile leaf-sheaths [after Luerssen (89); slightly less than nat. size]. B. Sporophyll bearing open sporangia (after Luerssen; slightly enlarged). C. Part of a transverse section (diagrammatic); v, vaUecular canals, e, en- dodermis, u, carinal canals (after Stock Photo
RMPG16RG–. Fossil plants : for students of botany and geology . Paleobotany. 246 PTERIDOPHYTA. [CH. main characters of the recent genus will enable the student to appreciate the points of difference and agreement between the extinct and present representatives of the Equisetales.. Fig. 52. Equisetum maximum Lam. A. Fertile shoot with strobilus and sterile leaf-sheaths [after Luerssen (89); slightly less than nat. size]. B. Sporophyll bearing open sporangia (after Luerssen; slightly enlarged). C. Part of a transverse section (diagrammatic); v, vaUecular canals, e, en- dodermis, u, carinal canals (after
An elementary course of infinitesimal calculus . follows that This is identical, except as to notation, with the formula(2) of Art. 152. 406 INFINITESIMAL CALCULUS. [CH. X 154 Osculating Circle. A slightly different way of treating the matter is basedon the notion of the osculating circle. If Q and R be twoneighbouring points of the curve, one on each side of P, weconsider the limiting value of the radius of the circle PQB,when Q and R are taken infinitely close to P. We can shew that if the curvature of the given curve becontinuous at P, this circle coincides in the limit with the circle of c Stock Photo
RM2AXJR1Y–An elementary course of infinitesimal calculus . follows that This is identical, except as to notation, with the formula(2) of Art. 152. 406 INFINITESIMAL CALCULUS. [CH. X 154 Osculating Circle. A slightly different way of treating the matter is basedon the notion of the osculating circle. If Q and R be twoneighbouring points of the curve, one on each side of P, weconsider the limiting value of the radius of the circle PQB,when Q and R are taken infinitely close to P. We can shew that if the curvature of the given curve becontinuous at P, this circle coincides in the limit with the circle of c
. Fossil plants : for students of botany and geology . Paleobotany. XL] GINKGOITES 17 The leaves of Ginlcgo digitata have a long slender petiole (fig. 634); the lamina is semiorbicular or obcuneate, entire, or more or less deeply divided into equal lobes, or irregularly divided into several. Fig. 635. Ginkgoites. (^ nat. size.) A. Ginkgoites adiantoides. Tertiary, Island of Mull. B. G. pluripartita, Wealden, North Germany (after Sohenk). C. G. digitata, Jurassic, Japan (after Yokoyama). D. G. digitata var. Huttoni, Jurassic, Australia (after Stirling). E. G. sibirica, Jurassic, Siberia (after Stock Photo
RMPG171A–. Fossil plants : for students of botany and geology . Paleobotany. XL] GINKGOITES 17 The leaves of Ginlcgo digitata have a long slender petiole (fig. 634); the lamina is semiorbicular or obcuneate, entire, or more or less deeply divided into equal lobes, or irregularly divided into several. Fig. 635. Ginkgoites. (^ nat. size.) A. Ginkgoites adiantoides. Tertiary, Island of Mull. B. G. pluripartita, Wealden, North Germany (after Sohenk). C. G. digitata, Jurassic, Japan (after Yokoyama). D. G. digitata var. Huttoni, Jurassic, Australia (after Stirling). E. G. sibirica, Jurassic, Siberia (after
An elementary course of infinitesimal calculus . Since these varia-tions of t are equal and opposite, and since the coordinatesof On are as a rule symmetric functions of ti, t^, the displace-ments of Oijwill in general be ultimately equal and opposite;that is, the triangle IfiiJ^i is ultimately isosceles, with itsequal angles infinitely small. It follows that when thepositions 1, 2, 3,... are taken closer and closer, the lengthof any finite portion of the polygon O^O^On... is ultimatelyequal to that of the corresponding arc of the space-centrode,and that any side of the polygon is ultimately a Stock Photo
RM2AXJHT8–An elementary course of infinitesimal calculus . Since these varia-tions of t are equal and opposite, and since the coordinatesof On are as a rule symmetric functions of ti, t^, the displace-ments of Oijwill in general be ultimately equal and opposite;that is, the triangle IfiiJ^i is ultimately isosceles, with itsequal angles infinitely small. It follows that when thepositions 1, 2, 3,... are taken closer and closer, the lengthof any finite portion of the polygon O^O^On... is ultimatelyequal to that of the corresponding arc of the space-centrode,and that any side of the polygon is ultimately a
. Fossil plants : for students of botany and geology . Paleobotany. xv] LEPIDODENDRON 121 An example of a large and partially decorticated stem is afforded by the specimen (16 x 7*5 cm.) shown in fig. 153. The irregularly ribbed surface is formed of rather thick-walled phelloderm, in which occur tangentially arranged rows . of secretory strands. The tapered form of the secondary cortex as it abuts internally on the primary cortex is shown very clearly in the drawing (cf fig. 151, C). The stele in this much older stem consists mainly of secondary wood.. Pig. 154. Lepidodendroti vasculare. Shoot Stock Photo
RMPG0D7P–. Fossil plants : for students of botany and geology . Paleobotany. xv] LEPIDODENDRON 121 An example of a large and partially decorticated stem is afforded by the specimen (16 x 7*5 cm.) shown in fig. 153. The irregularly ribbed surface is formed of rather thick-walled phelloderm, in which occur tangentially arranged rows . of secretory strands. The tapered form of the secondary cortex as it abuts internally on the primary cortex is shown very clearly in the drawing (cf fig. 151, C). The stele in this much older stem consists mainly of secondary wood.. Pig. 154. Lepidodendroti vasculare. Shoot
An elementary course of infinitesimal calculus . Fig. 73. transforms into an ellipse where a = aa, V = fia;. .(3),(4), 326 INFINITESIMAL CALCULUS. [CH. VIII and it is evident that by a proper choice of the ratios a, ^ acircle can be transformed into an ellipse of any given dimen-sions, and vice versd. Also since a system of parallel chords,and the diameter bisecting them, transform into a system ofparallel chords, and the diameter bisecting them, it is evidentthat perpendicular diameters of the circle transform intoconjugate diameters of the ellipse. Further, areas are altered by transformatio Stock Photo
RM2AXKA2X–An elementary course of infinitesimal calculus . Fig. 73. transforms into an ellipse where a = aa, V = fia;. .(3),(4), 326 INFINITESIMAL CALCULUS. [CH. VIII and it is evident that by a proper choice of the ratios a, ^ acircle can be transformed into an ellipse of any given dimen-sions, and vice versd. Also since a system of parallel chords,and the diameter bisecting them, transform into a system ofparallel chords, and the diameter bisecting them, it is evidentthat perpendicular diameters of the circle transform intoconjugate diameters of the ellipse. Further, areas are altered by transformatio
. Fossil plants : for students of botany and geology . Paleobotany. 216 LYCOPODIALES [CH. species, from the Coal-field of Valenciennes, the pedicel bore acicular leaves or bracts attached to the upper portion of leaf- cushions arranged in vertical series (fig. E). The cones reached a length of 16 cm. and a breadth of 2'5—5 cm.; the sporophylls are borne in alternating verticils with 8—10 in each whorl. Several megaspores (2 mm. in diameter) appear to have been produced in tetrads in each sporangium.. Fig. 201. Sigillariostrohus. A, C. Sigillariostrohus rhombibracteatus Kidst. (After Kidston.) Stock Photo
RMPG15XX–. Fossil plants : for students of botany and geology . Paleobotany. 216 LYCOPODIALES [CH. species, from the Coal-field of Valenciennes, the pedicel bore acicular leaves or bracts attached to the upper portion of leaf- cushions arranged in vertical series (fig. E). The cones reached a length of 16 cm. and a breadth of 2'5—5 cm.; the sporophylls are borne in alternating verticils with 8—10 in each whorl. Several megaspores (2 mm. in diameter) appear to have been produced in tetrads in each sporangium.. Fig. 201. Sigillariostrohus. A, C. Sigillariostrohus rhombibracteatus Kidst. (After Kidston.)
An elementary course of infinitesimal calculus . Fig. 133. 438 INFINITESIMAL CALCULUS. [CH. X of coordinates, the point of ultimate intersection with a consecutiveposition is at the foot Z of the perpendicular from the instan-taneous centre /. Now if AB = h, lOAB = ^, the coordinates of Z are given by X = BZcoa <^ = 57cos° ^ = A cos , y = AZ sin. ili = AI sin = k sin ) and the envelope oi AB is therefore the astroid - a? + y^ = k^ , see Fig. 134. Cf. Art. 138, Ex. 4. (3) •W;. Fig. 134. 164:. Application to Rolling Curves. When one plane curve rolls upon another, which isregarded as fixed, Stock Photo
RM2AXJM8A–An elementary course of infinitesimal calculus . Fig. 133. 438 INFINITESIMAL CALCULUS. [CH. X of coordinates, the point of ultimate intersection with a consecutiveposition is at the foot Z of the perpendicular from the instan-taneous centre /. Now if AB = h, lOAB = ^, the coordinates of Z are given by X = BZcoa <^ = 57cos° ^ = A cos , y = AZ sin. ili = AI sin = k sin ) and the envelope oi AB is therefore the astroid - a? + y^ = k^ , see Fig. 134. Cf. Art. 138, Ex. 4. (3) •W;. Fig. 134. 164:. Application to Rolling Curves. When one plane curve rolls upon another, which isregarded as fixed,
. Fossil plants : for students of botany and geology . Paleobotany. 492 CTCADOPHYTA [CH. zone is the secondary xylem the inner edge of which forms bays, and this is composed of alternating zones of vertical and horizontal tracheids (fig. 580, y-^—x^; fig. 581) with bordered, scalariform, pits on their walls traversed by medullary rays generally biseriate and from 4 to 30 cells deep. The disposition of the tracheids is such as to render transverse and radial longitudinal sections. Fig. .580. Golymbetes Edwardsi. Transverse section showing ten alternating zones of wood outside the perimedullary Stock Photo
RMPG06C5–. Fossil plants : for students of botany and geology . Paleobotany. 492 CTCADOPHYTA [CH. zone is the secondary xylem the inner edge of which forms bays, and this is composed of alternating zones of vertical and horizontal tracheids (fig. 580, y-^—x^; fig. 581) with bordered, scalariform, pits on their walls traversed by medullary rays generally biseriate and from 4 to 30 cells deep. The disposition of the tracheids is such as to render transverse and radial longitudinal sections. Fig. .580. Golymbetes Edwardsi. Transverse section showing ten alternating zones of wood outside the perimedullary
An elementary course of infinitesimal calculus . equal intervals of time. There are numerous optical and mechanical contrivances for producingthe curves. For a description of these, and for specimens of the curvesdescribed, we must refer to books on experimental Acoustics. 135-136] SPECIAL CURVES, 347 136. The Cycloid. The cycloid is the curve traced by a point on thecircumference of a circle which rolls in contact with a fixedstraight line. It evidently consists of an endless successionof exactly congruent portions, each of which represents acomplete revolution of the circle. The points (such Stock Photo
RM2AXK5E2–An elementary course of infinitesimal calculus . equal intervals of time. There are numerous optical and mechanical contrivances for producingthe curves. For a description of these, and for specimens of the curvesdescribed, we must refer to books on experimental Acoustics. 135-136] SPECIAL CURVES, 347 136. The Cycloid. The cycloid is the curve traced by a point on thecircumference of a circle which rolls in contact with a fixedstraight line. It evidently consists of an endless successionof exactly congruent portions, each of which represents acomplete revolution of the circle. The points (such
. Fossil plants : for students of botany and geology . Paleobotany. 18 GINKGOALES [CH. veins spread from the base of the lamina, the veins in the middle of the leaf being generally about 0-8—1 mm. apart. The stomata are practically confined to the lower surface of the lamina. The epidermal cells are polygonal and the walls slightly sinuous as in the recent species, and over the veins the cells are longer and narrower (fig. 636, A). Many of the larger epidermal cells have a cuticular ridge in the middle of the outer wall, represented in the figure by a black line. The stomata agree closely with Stock Photo
RMPG1715–. Fossil plants : for students of botany and geology . Paleobotany. 18 GINKGOALES [CH. veins spread from the base of the lamina, the veins in the middle of the leaf being generally about 0-8—1 mm. apart. The stomata are practically confined to the lower surface of the lamina. The epidermal cells are polygonal and the walls slightly sinuous as in the recent species, and over the veins the cells are longer and narrower (fig. 636, A). Many of the larger epidermal cells have a cuticular ridge in the middle of the outer wall, represented in the figure by a black line. The stomata agree closely with
An elementary course of infinitesimal calculus . contact withthe fixed line, and the point Z will move as if it were carried bythe small circle. Its locus is therefore a cycloid. Ex. 2. Similarly if a circle (A) roll on a fixed circle (B), theenvelope of any diameter of A is an epi- or hypo-cycloid whichwould be generated by the rolling of a circle of half the size of Aon the circumference of B. 165. Curvature of a Foint-Roulette. To investigate the curvature of any point P fixedrelatively to the rolling curve, let / be the point of contact,and let / be a consecutive point of contact, P the co Stock Photo
RM2AXJKNB–An elementary course of infinitesimal calculus . contact withthe fixed line, and the point Z will move as if it were carried bythe small circle. Its locus is therefore a cycloid. Ex. 2. Similarly if a circle (A) roll on a fixed circle (B), theenvelope of any diameter of A is an epi- or hypo-cycloid whichwould be generated by the rolling of a circle of half the size of Aon the circumference of B. 165. Curvature of a Foint-Roulette. To investigate the curvature of any point P fixedrelatively to the rolling curve, let / be the point of contact,and let / be a consecutive point of contact, P the co
. Fossil plants : for students of botany and geology . Paleobotany. xxxv] RHYNCHOGOlSrnjM 359 flora of Spitzbergen Nathorst^ discusses the morphological nature of Rhynchogonium seeds and describes additional specimens. Without the aid of petrified examples it is hardly possible to determine the true nature of the fossils.. Fig. 506. A, Holcospermum sulcatum. B, C, Codonospermum anomalum. D, Diplopterotesta spitzbergeneis (x 3). E, Gnetopsis eUiptica (oupule). F, Thysanotesta sagittula. G, Rhynchogonium costatum. H, Hexagonocarpus Noeggerathi. I, Boroviczia Karpinskii. K, Rhabdoapermum tunicatu Stock Photo
RMPG0CXB–. Fossil plants : for students of botany and geology . Paleobotany. xxxv] RHYNCHOGOlSrnjM 359 flora of Spitzbergen Nathorst^ discusses the morphological nature of Rhynchogonium seeds and describes additional specimens. Without the aid of petrified examples it is hardly possible to determine the true nature of the fossils.. Fig. 506. A, Holcospermum sulcatum. B, C, Codonospermum anomalum. D, Diplopterotesta spitzbergeneis (x 3). E, Gnetopsis eUiptica (oupule). F, Thysanotesta sagittula. G, Rhynchogonium costatum. H, Hexagonocarpus Noeggerathi. I, Boroviczia Karpinskii. K, Rhabdoapermum tunicatu
An elementary course of infinitesimal calculus . Fig. 66. lateral faces which are triangles or trapeziums. We may eveninclude the case where some or all of the lateral faces are curvedsurfaces (hyperbolic paraboloids) generated by straight hnesmoving parallel to the planes of the polygons, and each inter-secting two straight lines each of which joins a vertex of onepolygon to a vertex of the other (see Fig. 67). And since thenumber of sides in each polygon may be increased indefinitely,the rule will also apply to a solid bounded by any two plane 262 INFINITESIMAL CALCULnS. [CH. VII parallel fa Stock Photo
RM2AXKAKA–An elementary course of infinitesimal calculus . Fig. 66. lateral faces which are triangles or trapeziums. We may eveninclude the case where some or all of the lateral faces are curvedsurfaces (hyperbolic paraboloids) generated by straight hnesmoving parallel to the planes of the polygons, and each inter-secting two straight lines each of which joins a vertex of onepolygon to a vertex of the other (see Fig. 67). And since thenumber of sides in each polygon may be increased indefinitely,the rule will also apply to a solid bounded by any two plane 262 INFINITESIMAL CALCULnS. [CH. VII parallel fa
. Fossil plants : for students of botany and geology . Paleobotany. Fie. 624. Ctenis sp. Single pinna. (Manchester Museum, No. 53.'. Fig. 625. Ctenis sp. The epidermis of the pinna shown in fig. 624. A, surface- view; p, papillae; «, stomata. B, side-view. (Manchester Museum, No. 53.) an early paper as probably a new species of Anthrophyopsis, afforded good preparations of the cuticle (fig. 625) which revealed the existence of circular elevations on the outer walls of the epidermal cells: these may explain the nature of some at least 1 Seward (00) p 21; (00) B. p. 232, figs. 42, 43. 37—2. Plea Stock Photo
RMPG062K–. Fossil plants : for students of botany and geology . Paleobotany. Fie. 624. Ctenis sp. Single pinna. (Manchester Museum, No. 53.'. Fig. 625. Ctenis sp. The epidermis of the pinna shown in fig. 624. A, surface- view; p, papillae; «, stomata. B, side-view. (Manchester Museum, No. 53.) an early paper as probably a new species of Anthrophyopsis, afforded good preparations of the cuticle (fig. 625) which revealed the existence of circular elevations on the outer walls of the epidermal cells: these may explain the nature of some at least 1 Seward (00) p 21; (00) B. p. 232, figs. 42, 43. 37—2. Plea
An elementary course of infinitesimal calculus . 234 INFINITESIMAL CALCULUS. [CH. VI On the other hand, if 0 (x) be an odd function of x, so that we have <li{-x) = -{x). r <^{x)dx = 0.. J a •(5)..(6), since in the sum, of which the definite integral is the limit(Art. 86), the element cf) (x) Sx is cancelled by the oppositely-signed element 0 (— x) 8x.. Ex. 2. We have J—iir Jo whilst f sin 6 cos OdO^^O, since sin 0 changes sign with 0. For. similai^ reasons, if {a-x) = (x)dx=2l (j>(x)da; (8); Jo Jo whilst if ^(a-x) = -(j){x) (9), we have ( 6{x)dx = 0 (10). Jo As a particular case of ( Stock Photo
RM2AXKC9K–An elementary course of infinitesimal calculus . 234 INFINITESIMAL CALCULUS. [CH. VI On the other hand, if 0 (x) be an odd function of x, so that we have <li{-x) = -{x). r <^{x)dx = 0.. J a •(5)..(6), since in the sum, of which the definite integral is the limit(Art. 86), the element cf) (x) Sx is cancelled by the oppositely-signed element 0 (— x) 8x.. Ex. 2. We have J—iir Jo whilst f sin 6 cos OdO^^O, since sin 0 changes sign with 0. For. similai^ reasons, if {a-x) = (x)dx=2l (j>(x)da; (8); Jo Jo whilst if ^(a-x) = -(j){x) (9), we have ( 6{x)dx = 0 (10). Jo As a particular case of (
. Fossil plants : for students of botany and geology . Paleobotany. 226 THALLOPHTTA. [CH. they may at least be mentioned as possible but not certain Palaeozoic forms of Ghara or an allied genus.. d e Fig. 46. a. Chara Bleicheri Sa,Tp. x30. 6 and c. Devonian CAara ? sp. circa X 12. d and e. Chara Wrighti Forbes, circa x 12. 1. Chara Bleicheri, Saporta. Fig. 46, a. In this form the ' fruits' are minute and subspherical, "39â"44 mm. long, and â¢â 35â"40 mm. broad, showing in side view 5â6 slightly oblique spiral bands. Each spiral band bears a row of slightly project- ing tubercles Stock Photo
RMPG16WF–. Fossil plants : for students of botany and geology . Paleobotany. 226 THALLOPHTTA. [CH. they may at least be mentioned as possible but not certain Palaeozoic forms of Ghara or an allied genus.. d e Fig. 46. a. Chara Bleicheri Sa,Tp. x30. 6 and c. Devonian CAara ? sp. circa X 12. d and e. Chara Wrighti Forbes, circa x 12. 1. Chara Bleicheri, Saporta. Fig. 46, a. In this form the ' fruits' are minute and subspherical, "39â"44 mm. long, and â¢â 35â"40 mm. broad, showing in side view 5â6 slightly oblique spiral bands. Each spiral band bears a row of slightly project- ing tubercles
An elementary course of infinitesimal calculus . tly assumed, in the foregoing proof, that the areasare swept over always in the same direction. It is easy tosee, however, that the formula (1) will apply without anysuch restriction, provided areas be reckoned positive or nega-tive according as they are swept over towards the side of theline PQ on which ha is reckoned positive, or the reverse. 250 INFINITESIMAL CALCULUS. [CH. VII For example, the area swept over by a straight line whosemiddle point is fixed is on this reckoning zero. We will suppose, for definiteness, that So- is positive whent Stock Photo
RM2AXKATF–An elementary course of infinitesimal calculus . tly assumed, in the foregoing proof, that the areasare swept over always in the same direction. It is easy tosee, however, that the formula (1) will apply without anysuch restriction, provided areas be reckoned positive or nega-tive according as they are swept over towards the side of theline PQ on which ha is reckoned positive, or the reverse. 250 INFINITESIMAL CALCULUS. [CH. VII For example, the area swept over by a straight line whosemiddle point is fixed is on this reckoning zero. We will suppose, for definiteness, that So- is positive whent
. Fossil plants : for students of botany and geology . Paleobotany. 340 FILICALES [CH. 1841. Neuropteris Goeppertiana, Muenster, in Goeppert, Gattungen foss. Pflanz. Lief. 5 and 6, p. 104, Pis. viii.—x. 1856. Pecopieris Huttoniana, Zigno, Flor. foss. Oolit. Vol. I. p. 133. 1867. Acrostichites Qoeppertianus, Schenk, Foss. Flor. Grenzach. p. 44, PI. V. fig. 5, PI. VII. fig. 2.. Fig. 256. A. Cladophlebis denticulata. B. B'. Todites Williamsoni (fertile). C. T. Williamsoni (sterile pinna). D. Discopteris Rallii. E. E'. Kidstonia heracleensis. F. Todeopsis primaeva. G. Todites Williamsoni (sporangi Stock Photo
RMPG15FN–. Fossil plants : for students of botany and geology . Paleobotany. 340 FILICALES [CH. 1841. Neuropteris Goeppertiana, Muenster, in Goeppert, Gattungen foss. Pflanz. Lief. 5 and 6, p. 104, Pis. viii.—x. 1856. Pecopieris Huttoniana, Zigno, Flor. foss. Oolit. Vol. I. p. 133. 1867. Acrostichites Qoeppertianus, Schenk, Foss. Flor. Grenzach. p. 44, PI. V. fig. 5, PI. VII. fig. 2.. Fig. 256. A. Cladophlebis denticulata. B. B'. Todites Williamsoni (fertile). C. T. Williamsoni (sterile pinna). D. Discopteris Rallii. E. E'. Kidstonia heracleensis. F. Todeopsis primaeva. G. Todites Williamsoni (sporangi
An elementary course of infinitesimal calculus . be the angle which IZmakes with the normal to therolling curve at /, and // = Ss,we have ultimately Bs cos ^ ??. Fig. 139. GI.S0 (1). 444 INFINITESIMAL CALCULUS. [CH. X Hence, substituting the value of SO from Art. 164 (1),we have cos ^ 1 1 , . -W-R^R ^^^- The radius of curvature of the envelope is then given by p = GI + IZ (3)*. If, along the normal to the rolling curve at /, but in thedirection opposite to that chosen in the preceding Art., wemeasure off a length IK such that IKR^R ^ - and describe a circle on this line as diameter, it appe Stock Photo
RM2AXJKBH–An elementary course of infinitesimal calculus . be the angle which IZmakes with the normal to therolling curve at /, and // = Ss,we have ultimately Bs cos ^ ??. Fig. 139. GI.S0 (1). 444 INFINITESIMAL CALCULUS. [CH. X Hence, substituting the value of SO from Art. 164 (1),we have cos ^ 1 1 , . -W-R^R ^^^- The radius of curvature of the envelope is then given by p = GI + IZ (3)*. If, along the normal to the rolling curve at /, but in thedirection opposite to that chosen in the preceding Art., wemeasure off a length IK such that IKR^R ^ - and describe a circle on this line as diameter, it appe
. Fossil plants : for students of botany and geology . Paleobotany. 526. Fig. 597. Ptilophyllum boreale (Heer). Drawn from one of the specimens from the Lower Cretaceous beds at Ekkorfat, Greenland, figured by Heer. (Stockhobn Museum; nat. size and a, few pimiae enlarged.). Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original work.. Seward, A. C. (Albert Charles), 1863-1941. Cambridge : University Press Stock Photo
RMPG067R–. Fossil plants : for students of botany and geology . Paleobotany. 526. Fig. 597. Ptilophyllum boreale (Heer). Drawn from one of the specimens from the Lower Cretaceous beds at Ekkorfat, Greenland, figured by Heer. (Stockhobn Museum; nat. size and a, few pimiae enlarged.). Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original work.. Seward, A. C. (Albert Charles), 1863-1941. Cambridge : University Press
An elementary course of infinitesimal calculus . oefficients are chosen soas to make the function (3) and its first n — 1 derivatives coinciderespectively with f{x) and its first n—l derivatives for the par-ticular value x = 0. The result of this determination is, by Art.201, the function $„ (as). In the graphical representation, the parabolic curve y = *„ (a;)is determined so as to have contact of the (w-l)th order (seeArt. 206) with a given curve y=f{x) at the point a5= 0; and theproblem is, to find limits to the possible deviation of one curvefrom the other, as measured by the difierence of Stock Photo
RM2AXJCT7–An elementary course of infinitesimal calculus . oefficients are chosen soas to make the function (3) and its first n — 1 derivatives coinciderespectively with f{x) and its first n—l derivatives for the par-ticular value x = 0. The result of this determination is, by Art.201, the function $„ (as). In the graphical representation, the parabolic curve y = *„ (a;)is determined so as to have contact of the (w-l)th order (seeArt. 206) with a given curve y=f{x) at the point a5= 0; and theproblem is, to find limits to the possible deviation of one curvefrom the other, as measured by the difierence of
. Fossil plants : for students of botany and geology . Paleobotany. xv] LEPIDODENDRON 151 The longitudinal section (%. 168, C) shows the outer edge of the primary xylem, x, abutting on a band of dark and small- celled parenchyma which passes into the broad zone of secondary tissue, m, the inner region of which consists of fairly thick- walled elements in radial series passing externally into the thin-walled cells of the cambial region, a, on the inner edge of the secretory zone, sc. This section shows also the interruption. Fio. 171. Lepidodevdron faliginosum. (Prom sections in the Manchester Stock Photo
RMPG167R–. Fossil plants : for students of botany and geology . Paleobotany. xv] LEPIDODENDRON 151 The longitudinal section (%. 168, C) shows the outer edge of the primary xylem, x, abutting on a band of dark and small- celled parenchyma which passes into the broad zone of secondary tissue, m, the inner region of which consists of fairly thick- walled elements in radial series passing externally into the thin-walled cells of the cambial region, a, on the inner edge of the secretory zone, sc. This section shows also the interruption. Fio. 171. Lepidodevdron faliginosum. (Prom sections in the Manchester
An elementary course of infinitesimal calculus . Fig. 131. Hence the velocity of P is to that of Q as IP to IQ. Let PQ,produced if necessary, meet the perpendicular through 0 to theline of motion of P in the point E. Then if w be the angularvelocity of OQ, TP OR velocityof P = o).OGx^= co. 0^ x ^ = «,. OiJ...(l). Ex. 3. In an engine with an oscillating cylinder, the pistonrod, being connected directly with the crank OQ, has its directionalways passing through a fixed point G (on the pivot line of thecylinder). The instantaneous axis is now at the intersection of OQproduced with the perpendicul Stock Photo
RM2AXJMH7–An elementary course of infinitesimal calculus . Fig. 131. Hence the velocity of P is to that of Q as IP to IQ. Let PQ,produced if necessary, meet the perpendicular through 0 to theline of motion of P in the point E. Then if w be the angularvelocity of OQ, TP OR velocityof P = o).OGx^= co. 0^ x ^ = «,. OiJ...(l). Ex. 3. In an engine with an oscillating cylinder, the pistonrod, being connected directly with the crank OQ, has its directionalways passing through a fixed point G (on the pivot line of thecylinder). The instantaneous axis is now at the intersection of OQproduced with the perpendicul
. Fossil plants : for students of botany and geology . Paleobotany. 292 FILICALES [CH. the Malay region : M. pectinata grows in Western Borneo and m various localities in the Malay peninsula, while M. sarmentosa. MB. Fig. 227. Matoniapectinata. (^ nat. size.) M.S. has been found in one locality only; the latter species has recently been transferred to a new genus Phanerosorus, but in view of the practical identity in anatomical structure and. Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of Stock Photo
RMPG15P2–. Fossil plants : for students of botany and geology . Paleobotany. 292 FILICALES [CH. the Malay region : M. pectinata grows in Western Borneo and m various localities in the Malay peninsula, while M. sarmentosa. MB. Fig. 227. Matoniapectinata. (^ nat. size.) M.S. has been found in one locality only; the latter species has recently been transferred to a new genus Phanerosorus, but in view of the practical identity in anatomical structure and. Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of
An elementary course of infinitesimal calculus . Kg. 91. We proceed to notice in detail one or two of the caseswhich have specially important properties. Ex. 1. The cardioid. If in Art. 137 (3) we put 6 = a, we get x = 2a cos 6 + a cos 26, y = 2asm0+asin 29, whence a! + a = 2a(l+cos6)cos^, y = 2a (I + cos 6) sin 6...(S). This shews that the radius vector drawn from the point (- a, 0)as pole is given by r = 2o(l + cos6i) (4).. Fig. 92. 138] SPECIAL CURVES. 357 This is otherwise evident from Fig, 92, where AP = iAN=i{OI + AM). The corresponding trochoids are given by x = 2a cos 0 + k cos 2$, i/= Stock Photo
RM2AXK1KK–An elementary course of infinitesimal calculus . Kg. 91. We proceed to notice in detail one or two of the caseswhich have specially important properties. Ex. 1. The cardioid. If in Art. 137 (3) we put 6 = a, we get x = 2a cos 6 + a cos 26, y = 2asm0+asin 29, whence a! + a = 2a(l+cos6)cos^, y = 2a (I + cos 6) sin 6...(S). This shews that the radius vector drawn from the point (- a, 0)as pole is given by r = 2o(l + cos6i) (4).. Fig. 92. 138] SPECIAL CURVES. 357 This is otherwise evident from Fig, 92, where AP = iAN=i{OI + AM). The corresponding trochoids are given by x = 2a cos 0 + k cos 2$, i/=
. Fossil plants : for students of botany and geology . Paleobotany. 252 LYCOPODIALES [CH.. Fig. 212. Bothrodendron. A. Bothrodendron minutifolium, var. rotundatn Weiss. After Weiss and Sterzel. B. B. punctatum. After Zeiller. C. B. minutifoUmn. After Weiss and Sterzel. D. B. minutifolium. After Zeiller. E. Lepidostrobus Olryi. After Zeiller. F. Bothrodendron punctatum. After Zeiller. O, H. B. kiltorkense. G, after Nathorst; H, after Weiss and Sterzel.. Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and app Stock Photo
RMPG15WD–. Fossil plants : for students of botany and geology . Paleobotany. 252 LYCOPODIALES [CH.. Fig. 212. Bothrodendron. A. Bothrodendron minutifolium, var. rotundatn Weiss. After Weiss and Sterzel. B. B. punctatum. After Zeiller. C. B. minutifoUmn. After Weiss and Sterzel. D. B. minutifolium. After Zeiller. E. Lepidostrobus Olryi. After Zeiller. F. Bothrodendron punctatum. After Zeiller. O, H. B. kiltorkense. G, after Nathorst; H, after Weiss and Sterzel.. Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and app
An elementary course of infinitesimal calculus . able, and therefore, as regards the Integral Calculus, withproblems which depend, or can be made to depend, upon asingle integration. Multiple integrals occur however sofrequently, as a matter of notation, in the physical applica-tions of the subject, that it may be useful to give here a fewexplanations concerning them. We shall pass very lightlyover theoretical points; what is wanting in this respect maybe supplied by a proper adaptation of the method of Art. 87. * This Art. may be postponed. 114] GEOMETRICAL APPLICATIONS. 283 Let 0 be a contin Stock Photo
RM2AXKAEF–An elementary course of infinitesimal calculus . able, and therefore, as regards the Integral Calculus, withproblems which depend, or can be made to depend, upon asingle integration. Multiple integrals occur however sofrequently, as a matter of notation, in the physical applica-tions of the subject, that it may be useful to give here a fewexplanations concerning them. We shall pass very lightlyover theoretical points; what is wanting in this respect maybe supplied by a proper adaptation of the method of Art. 87. * This Art. may be postponed. 114] GEOMETRICAL APPLICATIONS. 283 Let 0 be a contin
. Fungoid and insect pests of the farm. Agricultural pests. 50 Mildews [CH.. Kg. 17. Erysiphe graminu. A, conidiophores bearing conidia; B, a spore case with appendages; C, a spore case which has burst showing the asci and ascospores. (Magnified.). Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original work.. Petherbridge, Frederick Robert. Cambridge : University Press Stock Photo
RMPG0RF0–. Fungoid and insect pests of the farm. Agricultural pests. 50 Mildews [CH.. Kg. 17. Erysiphe graminu. A, conidiophores bearing conidia; B, a spore case with appendages; C, a spore case which has burst showing the asci and ascospores. (Magnified.). Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original work.. Petherbridge, Frederick Robert. Cambridge : University Press
An elementary course of infinitesimal calculus . nts A, B, A, B areE, F, E, F, respectively. Ex. 3. To find the evolute of a cycloid. At any point P on the cycloid APB (Fig. 125), we have, byArt. 151, Ex. 2, p = 2PI (14). Let the axis AB be produced to D, so that BD = AB; andproduce TI to meet a parallel to BI, drawn through D, in /.If a circle be described on // as diameter, and PI be producedto meet the circumference in P, we have PI=PI, so that P isthe centre of curvature of the cycloid at P. And since the arcPi is equal to the arc TP, and therefore to BI or Dl, thelocus of P is evidently t Stock Photo
RM2AXJNG7–An elementary course of infinitesimal calculus . nts A, B, A, B areE, F, E, F, respectively. Ex. 3. To find the evolute of a cycloid. At any point P on the cycloid APB (Fig. 125), we have, byArt. 151, Ex. 2, p = 2PI (14). Let the axis AB be produced to D, so that BD = AB; andproduce TI to meet a parallel to BI, drawn through D, in /.If a circle be described on // as diameter, and PI be producedto meet the circumference in P, we have PI=PI, so that P isthe centre of curvature of the cycloid at P. And since the arcPi is equal to the arc TP, and therefore to BI or Dl, thelocus of P is evidently t
. Fossil plants : for students of botany and geology . Paleobotany. X] LEAVES OF CALAMITES. 381 epidermal layer with a limiting cuticle. Internal to this we have radially elongated parenchymatous cells forming a loose or spongy tissue, the cells being often separated by fairly large spaces. Pig. 86. A leaf of Calamites. 1. Transverse section; t, vascular bundle; a;, aheath of cells, x 35. 2. Vascular bundle consisting of a few small tracheids, t. 3. A tracheid and a few parenebymatous cells, the latter with nuclei. 4. A stoma; s, s, guard-cells. 5. PaUisade cells and intercellular spaces. From Stock Photo
RMPG0DBR–. Fossil plants : for students of botany and geology . Paleobotany. X] LEAVES OF CALAMITES. 381 epidermal layer with a limiting cuticle. Internal to this we have radially elongated parenchymatous cells forming a loose or spongy tissue, the cells being often separated by fairly large spaces. Pig. 86. A leaf of Calamites. 1. Transverse section; t, vascular bundle; a;, aheath of cells, x 35. 2. Vascular bundle consisting of a few small tracheids, t. 3. A tracheid and a few parenebymatous cells, the latter with nuclei. 4. A stoma; s, s, guard-cells. 5. PaUisade cells and intercellular spaces. From
An elementary course of infinitesimal calculus . 160, that the tangents fco the given curve are normalsto the locus of Q, so that this locus fulfils the above defini-tion of an involute. And, by varying the constant, weobtain a series of involutes of the same curve. As a concrete example we may imagine a string to be woundon a material arc of the given shape, being attached to a fixedpoint on it. The curve traced out by any point on the freeportion of the string will be an involute. This is in fact theorigin of the term. Ex. 1. The tractrix is an involute of the catenary; see Art.134. Ex. 2. I Stock Photo
RM2AXJN2J–An elementary course of infinitesimal calculus . 160, that the tangents fco the given curve are normalsto the locus of Q, so that this locus fulfils the above defini-tion of an involute. And, by varying the constant, weobtain a series of involutes of the same curve. As a concrete example we may imagine a string to be woundon a material arc of the given shape, being attached to a fixedpoint on it. The curve traced out by any point on the freeportion of the string will be an involute. This is in fact theorigin of the term. Ex. 1. The tractrix is an involute of the catenary; see Art.134. Ex. 2. I
. Fossil plants : for students of botany and geology . Paleobotany. xxxix] TSriLSSOMA 567. Fig. 619. A, seed {NiUsonia'!); B, NiUsonia pterophylloidea; C, D, Nilssonia brevis; E, N. polymorpha. (A—D f nat. size; E, slightly enlarged; after Nathorst.). Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original work.. Seward, A. C. (Albert Charles), 1863-1941. Cambridge : University Press Stock Photo
RMPG063M–. Fossil plants : for students of botany and geology . Paleobotany. xxxix] TSriLSSOMA 567. Fig. 619. A, seed {NiUsonia'!); B, NiUsonia pterophylloidea; C, D, Nilssonia brevis; E, N. polymorpha. (A—D f nat. size; E, slightly enlarged; after Nathorst.). Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original work.. Seward, A. C. (Albert Charles), 1863-1941. Cambridge : University Press
An elementary course of infinitesimal calculus . consecutive normals, the evolute is alsothe envelope of the normals to the given curve. Hence thenormals to the original curve are tangents to the evolute*. Ex. 1. In the parabola y^ = ^ax (1), we have a; = as cot^i/r, 2/ = 2ascoti/f (2), and (by Art. 151, Ex. 4) p = -2a/sm^f (3). The coordinates of the centre of curvature are therefore i = x- pahxtj/ = 3x + 2a, ... ?q = y + p cos i/f = — y^/ia^) Hence rf = yjUa^ = ia^ja = A (f - ^af/a. The evolute is therefore the semi-cubical parabola af = .^{x-2af (5). Otherwise : it is shewn in books on Co Stock Photo
RM2AXJPAB–An elementary course of infinitesimal calculus . consecutive normals, the evolute is alsothe envelope of the normals to the given curve. Hence thenormals to the original curve are tangents to the evolute*. Ex. 1. In the parabola y^ = ^ax (1), we have a; = as cot^i/r, 2/ = 2ascoti/f (2), and (by Art. 151, Ex. 4) p = -2a/sm^f (3). The coordinates of the centre of curvature are therefore i = x- pahxtj/ = 3x + 2a, ... ?q = y + p cos i/f = — y^/ia^) Hence rf = yjUa^ = ia^ja = A (f - ^af/a. The evolute is therefore the semi-cubical parabola af = .^{x-2af (5). Otherwise : it is shewn in books on Co
. Fossil plants : for students of botany and geology . Paleobotany. 98 LTCOPODIALES [CH. and slender acicular needles very similar to those of Firms longifolium; in L, Stemhergii (fig. 141) they are much broader and shorter. In external form as in internal structure it is often impossible to distinguish between the leaves of Lepidodendron and Sigillaria, The distinguishing features enumerated by the late M. Kenault cannot be employed, with any great degree of confidence, as diagnostic characters. In transverse section the lamina of a Lepidodendron leaf presents the same appearance as that of t Stock Photo
RMPG0DB7–. Fossil plants : for students of botany and geology . Paleobotany. 98 LTCOPODIALES [CH. and slender acicular needles very similar to those of Firms longifolium; in L, Stemhergii (fig. 141) they are much broader and shorter. In external form as in internal structure it is often impossible to distinguish between the leaves of Lepidodendron and Sigillaria, The distinguishing features enumerated by the late M. Kenault cannot be employed, with any great degree of confidence, as diagnostic characters. In transverse section the lamina of a Lepidodendron leaf presents the same appearance as that of t
An elementary course of infinitesimal calculus . nd figure. Hence thearea of the hyperbolic sector AOP is lPN.0N-a.rQ3.PAN=lu (10). We have here an analogy between the amplitude (m) of thehyperbolic functions cosh it, sinh u, &c., and the amplitude {&)of the circular functions cos 0, sin 6, &c.; viz. the independentvariable in each case represents twice the sectorial area AOPcorresponding to the point P whose coordinates are (cosh it,sinh It), or (cos B, sin 6), respectively. In the case of the general hyperbola S-F = ^ (^^) the coordinates of any point on the positive branch may berepresented Stock Photo
RM2AXKB2E–An elementary course of infinitesimal calculus . nd figure. Hence thearea of the hyperbolic sector AOP is lPN.0N-a.rQ3.PAN=lu (10). We have here an analogy between the amplitude (m) of thehyperbolic functions cosh it, sinh u, &c., and the amplitude {&)of the circular functions cos 0, sin 6, &c.; viz. the independentvariable in each case represents twice the sectorial area AOPcorresponding to the point P whose coordinates are (cosh it,sinh It), or (cos B, sin 6), respectively. In the case of the general hyperbola S-F = ^ (^^) the coordinates of any point on the positive branch may berepresented
. Fossil plants : for students of botany and geology . Paleobotany. 146 CONIFEEALES (RECENT) [CH. CUPBESSINEAE. The absence of resin-canals in the xylem is a feature shared by other families; but in the occurrence of xylem-parenchyma in different regions of the wood the Cupressineae differ as a rule from the Abie- tineae, though this is not a constant distinguishing character. The pits in the field vary from 1 to 6 or 8 in some genera, e.g., Taxodium and Glypto- strobus^: Gothan^ applies the term Cupressoid to medullary-ray pits character-. FiG. 698. Araucariu exceha. A, Mature form. B, Seedli Stock Photo
RMPG0DAT–. Fossil plants : for students of botany and geology . Paleobotany. 146 CONIFEEALES (RECENT) [CH. CUPBESSINEAE. The absence of resin-canals in the xylem is a feature shared by other families; but in the occurrence of xylem-parenchyma in different regions of the wood the Cupressineae differ as a rule from the Abie- tineae, though this is not a constant distinguishing character. The pits in the field vary from 1 to 6 or 8 in some genera, e.g., Taxodium and Glypto- strobus^: Gothan^ applies the term Cupressoid to medullary-ray pits character-. FiG. 698. Araucariu exceha. A, Mature form. B, Seedli
An elementary course of infinitesimal calculus . Fig. 74. of equation being (10). Finally, we have a single branch passingoutside the loop in the last case; the equation is again of thetype (9)*. * The curves in the : gure have been traced from the equation where C= - 2,0,2, 4, 6. The relation between them is most easily conceivedby regarding them as successive contour-lines of a surface (Art. 43), as inthe neighbourhood of a pinnacle on a mountain side. 336 INFINITESIMAL CALCULUS. [CH. IX (e) In the very special case where all three quantities a,)3, y, in (8), coincide, so that ay* = {x — a) Stock Photo
RM2AXK9C9–An elementary course of infinitesimal calculus . Fig. 74. of equation being (10). Finally, we have a single branch passingoutside the loop in the last case; the equation is again of thetype (9)*. * The curves in the : gure have been traced from the equation where C= - 2,0,2, 4, 6. The relation between them is most easily conceivedby regarding them as successive contour-lines of a surface (Art. 43), as inthe neighbourhood of a pinnacle on a mountain side. 336 INFINITESIMAL CALCULUS. [CH. IX (e) In the very special case where all three quantities a,)3, y, in (8), coincide, so that ay* = {x — a)
. Fossil plants : for students of botany and geology . Paleobotany. 372 FILICALES [CH. fossil fern-stems with the habit and, in the main, the structural features of recent tree-ferns. Persistent leaf-bases and sinuous adventitious roots cover the surface of the stems: the vascular system is of the dictyostelic type characteristic of Gyathea (fig. 240, p. 313) and Alsophila. It is by the pattern formed by the vascular tissue on the exposed surface of the leaf-bases that Protopteris is most readily recognised: the leaf-trace has a horse- shoe form with the ends curled inwards and the sides more Stock Photo
RMPG15CA–. Fossil plants : for students of botany and geology . Paleobotany. 372 FILICALES [CH. fossil fern-stems with the habit and, in the main, the structural features of recent tree-ferns. Persistent leaf-bases and sinuous adventitious roots cover the surface of the stems: the vascular system is of the dictyostelic type characteristic of Gyathea (fig. 240, p. 313) and Alsophila. It is by the pattern formed by the vascular tissue on the exposed surface of the leaf-bases that Protopteris is most readily recognised: the leaf-trace has a horse- shoe form with the ends curled inwards and the sides more
Travels of a consular officer in eastern Tibet : together with a history of the relations between China, Tibet and India . 103O0 y Jyara^n^ , ikVanqhw /iVp <wba«*i lZOOO •x3 Bol ^•^«»^amAa lax, 30^508GAmba^^^^Gtun (Angti,) ja2oc ^ tefoo kGa7n,Jda i ^ ^. V o HartgcUi Gomha. -^- —- —- Ncbdiconq La ^ nny^tJL LaNyashiy^ /Be fPaiyol^ Gomb ^ixtsss^^ 1300V - JirtAo Gornhou i s o Lofn^ *w Loj 15500^— i , ,- J-)rawa.)yamaTjn. i4goo^ Goixjo D;qot(^ (Kungchuch)lyrondrf-n. Gornba, w - :^ moo (;„,„i^44.^r».a,J8fcc. V^ . ; . D R . (/ rcLS sy BohvAoj Ganibi (Pangtcu). iLoa PiLchiLr Stock Photo
RM2AN6J14–Travels of a consular officer in eastern Tibet : together with a history of the relations between China, Tibet and India . 103O0 y Jyara^n^ , ikVanqhw /iVp <wba«*i lZOOO •x3 Bol ^•^«»^amAa lax, 30^508GAmba^^^^Gtun (Angti,) ja2oc ^ tefoo kGa7n,Jda i ^ ^. V o HartgcUi Gomha. -^- —- —- Ncbdiconq La ^ nny^tJL LaNyashiy^ /Be fPaiyol^ Gomb ^ixtsss^^ 1300V - JirtAo Gornhou i s o Lofn^ *w Loj 15500^— i , ,- J-)rawa.)yamaTjn. i4goo^ Goixjo D;qot(^ (Kungchuch)lyrondrf-n. Gornba, w - :^ moo (;„,„i^44.^r».a,J8fcc. V^ . ; . D R . (/ rcLS sy BohvAoj Ganibi (Pangtcu). iLoa PiLchiLr
. Fossil plants : for students of botany and geology . Paleobotany. xxv] ANKYROPTERIS 453 internally of parenchyma which becomes thicker-walled towards the periphery and bears multicellular epidermal hairs. A leaf- trace is detached in the form of a triangular strand and is formed by the tangential extension of the distal end of an arm of the stele. The trace, on its way through the cortex, divides into two; the outer branch gradually changes from a slightly curved band to an H-shaped meristele; the inner branch, which supplied an axillary shoot, is similar to the stele of the. Please note tha Stock Photo
RMPG1513–. Fossil plants : for students of botany and geology . Paleobotany. xxv] ANKYROPTERIS 453 internally of parenchyma which becomes thicker-walled towards the periphery and bears multicellular epidermal hairs. A leaf- trace is detached in the form of a triangular strand and is formed by the tangential extension of the distal end of an arm of the stele. The trace, on its way through the cortex, divides into two; the outer branch gradually changes from a slightly curved band to an H-shaped meristele; the inner branch, which supplied an axillary shoot, is similar to the stele of the. Please note tha
An elementary course of infinitesimal calculus . tuitively from the definition of Art. 86. For example, ra ra I (f)(x)dx = I ^{a — x)dx (1). This is proved by writing x = a — x, dx= — dx, 95-96] DEFINITE INTEGRALS. 233 the new limits of integration being « =a, x = 0, correspond-ing to x = 0,x = a, respectively. Thus ra ro ra I (a — ») dso = j ^(a — x) dx, Jo J a Jo the accent being dropped in the end, as no longer necessary. This process is equivalent to transferring the origin tothe point x = a, and reversing the direction of the axis of x.The areas represented by the integrals in (1) are thu Stock Photo
RM2AXKCM5–An elementary course of infinitesimal calculus . tuitively from the definition of Art. 86. For example, ra ra I (f)(x)dx = I ^{a — x)dx (1). This is proved by writing x = a — x, dx= — dx, 95-96] DEFINITE INTEGRALS. 233 the new limits of integration being « =a, x = 0, correspond-ing to x = 0,x = a, respectively. Thus ra ro ra I (a — ») dso = j ^(a — x) dx, Jo J a Jo the accent being dropped in the end, as no longer necessary. This process is equivalent to transferring the origin tothe point x = a, and reversing the direction of the axis of x.The areas represented by the integrals in (1) are thu
. Fossil plants : for students of botany and geology . Paleobotany. XXXIX] PTEEOPHYLLTJM 551 species is, however, probably a Nilssonia. The stoma represented in fig. 611, A, is from the lower epidermis of Pterophyllum (Anomozamites) Nilssoni (Phill.)^ Pterophyllum extends from Upper Carboniferous to Lower Cretaceous strata and is especially characteristic of Keuper and Rhaetic floras. A possible representative of the sub-genus Anomozamites is described by Ettingshausen from Tertiary beds in New South "Wales as Anomozamites Mmlleri^. AT'. A ^ B Fig. 611. A, Pierophyllum Nilssoni; B, Nilsso Stock Photo
RMPG0658–. Fossil plants : for students of botany and geology . Paleobotany. XXXIX] PTEEOPHYLLTJM 551 species is, however, probably a Nilssonia. The stoma represented in fig. 611, A, is from the lower epidermis of Pterophyllum (Anomozamites) Nilssoni (Phill.)^ Pterophyllum extends from Upper Carboniferous to Lower Cretaceous strata and is especially characteristic of Keuper and Rhaetic floras. A possible representative of the sub-genus Anomozamites is described by Ettingshausen from Tertiary beds in New South "Wales as Anomozamites Mmlleri^. AT'. A ^ B Fig. 611. A, Pierophyllum Nilssoni; B, Nilsso
An elementary course of infinitesimal calculus . = 2a (6), we find cos^ = cos^ (7), or, the focal distances make equal angles with the curve onopposite sides. Ex. 2. To find the form which a reflecting or refractingsurface must have in order that incident rays whose directionspass through a fixed point S may be reflected or refracted indirections passing through a fixed point iS. The case of reflection is merely the converse of Ex. 1. Thesurface must have the form generated by the revolution of anellipse or hyperbola about the line joining the foci (S, S). 25—2 388 INFINITESIMAL CALCULUS. [OH. Stock Photo
RM2AXJT0H–An elementary course of infinitesimal calculus . = 2a (6), we find cos^ = cos^ (7), or, the focal distances make equal angles with the curve onopposite sides. Ex. 2. To find the form which a reflecting or refractingsurface must have in order that incident rays whose directionspass through a fixed point S may be reflected or refracted indirections passing through a fixed point iS. The case of reflection is merely the converse of Ex. 1. Thesurface must have the form generated by the revolution of anellipse or hyperbola about the line joining the foci (S, S). 25—2 388 INFINITESIMAL CALCULUS. [OH.
. Fossil plants : for students of botany and geology . Paleobotany. XXVIl] WEICHSELIA 495 Frond bipinnate, rachis broad ; pinnae very long, of uniform breadth and with prominent axes ; pinnules crowded, entire, with obtuse apex, usually oblong but more or less triangular or rounded towards the distal. B il l^.^ ^AX. Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original work.. Seward, A. C. (Albert Charles), 1863-1941. Cambridge : Univ Stock Photo
RMPG14NB–. Fossil plants : for students of botany and geology . Paleobotany. XXVIl] WEICHSELIA 495 Frond bipinnate, rachis broad ; pinnae very long, of uniform breadth and with prominent axes ; pinnules crowded, entire, with obtuse apex, usually oblong but more or less triangular or rounded towards the distal. B il l^.^ ^AX. Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original work.. Seward, A. C. (Albert Charles), 1863-1941. Cambridge : Univ
An elementary course of infinitesimal calculus . ng from 0 to any fixed value shortof o. As a matter of fact it will be uniformly convergent up to a;=a,inclusively, but this cannot be established by the above method. + By this notation it is meant that x may range from a to 6 inclusively. L. 35 546 INFINITESIMAL CALCULUS. [CH. Xlll The question may be illustrated graphically* by drawing thecurve y = S (x), and also the approximation-curves y = S„ (a;),for ?i= 1, 2, 3,...t. If the series be uniformly convergent, then,however small the value of o-, from some finite value of n onwardsthe approxim Stock Photo
RM2AXJEDD–An elementary course of infinitesimal calculus . ng from 0 to any fixed value shortof o. As a matter of fact it will be uniformly convergent up to a;=a,inclusively, but this cannot be established by the above method. + By this notation it is meant that x may range from a to 6 inclusively. L. 35 546 INFINITESIMAL CALCULUS. [CH. Xlll The question may be illustrated graphically* by drawing thecurve y = S (x), and also the approximation-curves y = S„ (a;),for ?i= 1, 2, 3,...t. If the series be uniformly convergent, then,however small the value of o-, from some finite value of n onwardsthe approxim
. Fossil plants : for students of botany and geology . Paleobotany. XX] CYATHEACEAE 295 pound, but Gyathea sinnata Hook, a rare Ceylon species, bears simple narrow linear leaves. This family includes, with few exceptions, all the tree ferns ^. The sori of Dicksonia are enclosed in a two-valved indusium (fig. 229, F, G); in the species represented in fig. 230 the fertile segments, which terminate in cup-like indusia, are characterised by the absence of a lamina and closely resemble those of Thyrsopteris (fig. 229, A).. Fio. 230. Dicksonia Bertercana Hook. Fertile aud sterile pinnae. (Nat. size. Stock Photo
RMPG15NF–. Fossil plants : for students of botany and geology . Paleobotany. XX] CYATHEACEAE 295 pound, but Gyathea sinnata Hook, a rare Ceylon species, bears simple narrow linear leaves. This family includes, with few exceptions, all the tree ferns ^. The sori of Dicksonia are enclosed in a two-valved indusium (fig. 229, F, G); in the species represented in fig. 230 the fertile segments, which terminate in cup-like indusia, are characterised by the absence of a lamina and closely resemble those of Thyrsopteris (fig. 229, A).. Fio. 230. Dicksonia Bertercana Hook. Fertile aud sterile pinnae. (Nat. size.
An elementary course of infinitesimal calculus . C Fig. 123. Ex. 2. The normal at any point of the ellipse a: = acos^, y = 6sin<^ (8) i, - J^ = „» 6^ (9). COS0 sin^ DiflFerentiating with respect to say (10). cos<^ sin^ ^ Substituting in (9), we have =a-h (11). Hence the coordinates of the centre of curvature are x = cos<^, y = 7— sin<^ (12); Of o and the evolute is (ax)i + {by)i = {a^-b)i (13). 159] CURVATURE. 423 This curve, which may be obtained by homogeneous strain fromthe astroid, is shewn in Fig. 124.. Pig. 124. The centres of curvature at the points A, B, A, B areE, F, E, Stock Photo
RM2AXJNTY–An elementary course of infinitesimal calculus . C Fig. 123. Ex. 2. The normal at any point of the ellipse a: = acos^, y = 6sin<^ (8) i, - J^ = „» 6^ (9). COS0 sin^ DiflFerentiating with respect to say (10). cos<^ sin^ ^ Substituting in (9), we have =a-h (11). Hence the coordinates of the centre of curvature are x = cos<^, y = 7— sin<^ (12); Of o and the evolute is (ax)i + {by)i = {a^-b)i (13). 159] CURVATURE. 423 This curve, which may be obtained by homogeneous strain fromthe astroid, is shewn in Fig. 124.. Pig. 124. The centres of curvature at the points A, B, A, B areE, F, E,