# . Differential and integral calculus. Fig. 24. Thus, Figs. 24, the curve MJV is convex, and ST is concaveto the pole O. Direction of Curvature Points of Inflexion 177 130. Investigation for Direction of Curvature. Let r =f{6) be the equation of either of the curves MN, ST, Figs. 24,— MN and ST being any two curves referred to polar co-ordinates. Let PB, P B, be two tangents drawn at any twopoints P, P; and let OB, OB be perpendiculars let fall fromthe pole on these tangents. From Fig. 24 (a) we see that asr {OP) increases/ {OB) decreases, and from Fig. 24 {b) thatas r {OP) increases/ {OB) incr

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. Differential and integral calculus. Fig. 24. Thus, Figs. 24, the curve MJV is convex, and ST is concaveto the pole O. Direction of Curvature Points of Inflexion 177 130. Investigation for Direction of Curvature. Let r =f{6) be the equation of either of the curves MN, ST, Figs. 24, — MN and ST being any two curves referred to polar co-ordinates. Let PB, P B, be two tangents drawn at any twopoints P, P; and let OB, OB be perpendiculars let fall fromthe pole on these tangents. From Fig. 24 (a) we see that asr {OP) increases/ {OB) decreases, and from Fig. 24 {b) thatas r {OP) increases/ {OB) increases ; hence, in either case, p = F{r), and since p is a decreasing function of r when the curve isconvex and an increasing function of r when it is concave, we have dp -f o, dr according as the curve is convex or concave to the pole.From § 78, we have p = M«y hence to investigate any curve r —/{\$) for direction of curva- drture at a given point, we first obtain — from the equation of the curve and substitute

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