. On Cauchy's modulus surfaces. .4x^Z4>r2x^- 4x 4 2Z^-4Zx^ 0 Or, x%% 2xzf2x^z f x^- 2x^4 z^-Hi 4^ = 0 .Ox (xz4x4z-l) = 0(xzfxi^z-1) = 0Z(xfl)+ (x-1) = 0, 9) Z = - x-1 for the equation of tl^e hyperbola. For ^ the intersection ofthe surfficc and the xz plane is bhtained. The quartic degeneratesinto the cubic(10) Z(xfl)^- (x-1)^ 0 and the lins Z =| 1 . Figure = 2 uniis CO XcJxi5 3 34IV. (1) W = ufiv = (x+i^f - 1 = x^-j- nixy - 3xy - ly - 1u = x^- 3xy^- 1T = 3x y - y 3.^ u% v^ (x^- 3xy^- 1) V (3x^y-/) = x+ 9xV-f 1 - Sx^y*- 2x^f 6x y^^f 9x^y% y^- fix^yZ = x^ 3x%*- 2x^ 3xyf 6xyN 1 4 y^
. On Cauchy's modulus surfaces. .4x^Z4>r2x^- 4x 4 2Z^-4Zx^ 0 Or, x%% 2xzf2x^z f x^- 2x^4 z^-Hi 4^ = 0 .Ox (xz4x4z-l) = 0(xzfxi^z-1) = 0Z(xfl)+ (x-1) = 0, 9) Z = - x-1 for the equation of tl^e hyperbola. For ^ the intersection ofthe surfficc and the xz plane is bhtained. The quartic degeneratesinto the cubic(10) Z(xfl)^- (x-1)^ 0 and the lins Z =| 1 . Figure = 2 uniis CO XcJxi5 3 34IV. (1) W = ufiv = (x+i^f - 1 = x^-j- nixy - 3xy - ly - 1u = x^- 3xy^- 1T = 3x y - y 3.^ u% v^ (x^- 3xy^- 1) V (3x^y-/) = x+ 9xV-f 1 - Sx^y*- 2x^f 6x y^^f 9x^y% y^- fix^yZ = x^ 3x%*- 2x^ 3xyf 6xyN 1 4 y^is the equation of the surface. i As foE W = -1, three roots of the equation in Z,z^- 1 - W = 0 become equal to zero, then to the unit circle / Wj^^ 1 thru -1 in the W-plane corresponds in the xy-plane a curve which, according to the g:eneral theory has a triple-point at the origin and three hraneh o es at ecTual angles of 60. To find this curve of intersection of the surface and a planeparallel to the xy plane at a distance 1. Solve simultaneously l)z = 1 and 2) Z = xS 3x*^y^- 2x^f 3x^y^V 6x y% 1 f y,which gives: 3) f(x,y) = 3xV^- 2x^4 3x^/j-6xj\ 0. This has no real asymptotes, as we see hy suhstituting y «Vx4h, and solving. f•fell. For lim /f(x,jO_i we get lim It-=00 or writing lim/yi = m (Um^)^. Consequently, (m-ti, x = , y^^c^) satisfy the
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