. On Cauchy's modulus surfaces. W = az f h, similitude A 4- rotation^ V translation h. W = C a, f ia,^)^ ( X iy) f h, f ih^ = a^x ^ h^ - a^y ±(a^x - y ^ h^V , = ( a,x V Ti, - a^y) ^ ( a^x v a,y | b^) = a^x*V ^jt^*^ ^y*- Sa, a^xy y a^xy -4- 2a,b^x f- Sa^b^^x 2a.^ bj^y - ^a^Lb, y b, b^ = (a^^a^^)xv (a,S a^)yV^( ^ ^ a^^ Jx-V 2(a, b^- a^b, )y f ^?t^ . Put Z = k, to determine the nature of the curves of intersection of planes parallel to the xy plane with the surface. Then we get (x^f y**) V g(a. b, -V a^bQx, 2(a,ba^- a,.b|)y = K- T>*-f a^- a^^ a^^f a]; Or, completing the square; A * a
![. On Cauchy's modulus surfaces. W = az f h, similitude A 4- rotation^ V translation h. W = C a, f ia,^)^ ( X iy) f h, f ih^ = a^x ^ h^ - a^y ±(a^x - y ^ h^V , = ( a,x V Ti, - a^y) ^ ( a^x v a,y | b^) = a^x*V ^jt^*^ ^y*~- Sa, a^xy y a^xy -4- 2a,b^x f- Sa^b^^x 2a.^ bj^y - ^a^Lb, y b, b^ = (a^^a^^)xv (a,S a^)yV^( ^ ^ a^^ Jx-V 2(a, b^- a^b, )y f ^?t^ . Put Z = k, to determine the nature of the curves of intersection of planes parallel to the xy plane with the surface. Then we get (x^f y**) V g(a. b, -V a^bQx, 2(a,ba^- a,.b|)y = K- T>*-f a^- a^^ a^^f a]; Or, completing the square; A * a Stock Photo](https://c8.alamy.com/comp/2AG8CXF/on-cauchys-modulus-surfaces-w-=-az-f-h-similitude-a-4-rotation-v-translation-h-w-=-c-a-f-ia-x-iy-f-h-f-ih-=-ax-h-ay-ax-y-hv-=-ax-v-ti-ay-ax-v-ay-b-=-axv-jt-y-sa-axy-y-axy-4-2abx-f-sabx-2a-bjy-alb-y-b-b-=-aaxv-as-ayv-a-jx-v-2a-b-ab-y-f-t-put-z-=-k-to-determine-the-nature-of-the-curves-of-intersection-of-planes-parallel-to-the-xy-plane-with-the-surface-then-we-get-xf-y-v-ga-b-v-abqx-2aba-aby-=-k-tgt-f-a-a-af-a-or-completing-the-square-a-a-2AG8CXF.jpg)
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. On Cauchy's modulus surfaces. W = az f h, similitude A 4- rotation^ V translation h. W = C a, f ia, ^)^ ( X iy) f h, f ih^ = a^x ^ h^ - a^y ±(a^x - y ^ h^V , = ( a, x V Ti, - a^y) ^ ( a^x v a, y | b^) = a^x*V ^jt^*^ ^y*~- Sa, a^xy y a^xy -4- 2a, b^x f- Sa^b^^x 2a.^ bj^y - ^a^Lb, y b, b^ = (a^^a^^)xv (a, S a^)yV^( ^ ^ a^^ Jx-V 2(a, b^- a^b, )y f ^?t^ . Put Z = k, to determine the nature of the curves of intersection of planes parallel to the xy plane with the surface. Then we get (x^f y**) V g(a. b, -V a^bQx, 2(a, ba^- a, .b|)y = K- T>*-f a^- a^^ a^^f a]; Or, completing the square; A * a, b, V a^hA±/y a, b^-ab - K - b, ^f b^, (a, b, % h>f (a.h^I a.-fa- I I Z^^^^Ty KfnTa^ (a, -+ a^ )I = K. Therefore tie sections made hy plares parallel to the xy planeare circles and the surface is a paraboloid of revolution, with theaxis of revolution perpendicular to the xy plane. To find the nature of the sections made by planes parallel tothe xz plane, let y = k and substitute this value for y in theequation of the surface. This then becomes ^ /x ^ a^b^a^r - Z = - a.T^^- and the sections are seen to be^parSolas. Semilarly the^krctions made by plaoes parallelto yz plane- found by puttinp^ x = k, are seen to be parabolas. To find whether or not the vertex of the paraboloid lies in thexy plane, put Z = 0 ami investigate the nature of the curve of inter-section. This gives (x f a. b. a^b^^^^ - a^I^aJiiJlL^v) = ^ V a - 7 V a*t ^ ^ , which IS a null circle, or the point/- a, b^ a^b ^. a, h^- n a i . I aj^ ii^ a^+ J Therefore the vertex of the paraboloid of revolution lies in the xy (- a^b, i a^J^^ a, b^- b, a^] which in the superposed complex plane is equivalent to W = 0, orZ = - b/a .