File:On Cauchy's modulus surfaces (1913) (14597542739).jpg

Identifier: oncauchysmodulus00grim (find matches)
Title: On Cauchy's modulus surfaces
Year: 1913 (1910s)
Authors: Grimes, Ruby Mabel
Subjects: Cauchy, Augustin Louis, Baron, 1789-1857 Surfaces, Algebraic Theses
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Contributing Library: University of Illinois Urbana-Champaign
Digitizing Sponsor: University of Illinois Urbana-Champaign

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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
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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 .

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