. On Cauchy's modulus surfaces. 0x°i. e,x \ 9 i y*= 8 (xt3f t y-= 8, Therefore the section is a circle, of radius 8 and center at (-3,0).Let Z = l/8. Then we havex^i 2x + 1 i y^= Sx^- 16x 8 ^ 8y^, or,-7x^-^ 18x - 7-7j^= 0x^- 18 x V 1 •{ y = ^ (x*-- 18 x f 81) \ y^=(4 7 49 49 (x-9/7)*-t y =_4. 49 Therefore the section is a circle with radius 2/7 and center at ( 9/7, 0). Similarly, putting Z = l/4, we get a circle with radius4/3 and center at (5/3, O). So also with X = 3/4, the section is acircle, with center at radius = 148. For the general case then, i>^ Z = k,X- 4 2kx f k H ky^- x*-i 2x -1
. On Cauchy's modulus surfaces. 0x°i. e,x \ 9 i y*= 8 (xt3f t y-= 8, Therefore the section is a circle, of radius 8 and center at (-3,0).Let Z = l/8. Then we havex^i 2x + 1 i y^= Sx^- 16x 8 ^ 8y^, or,-7x^-^ 18x - 7-7j^= 0x^- 18 x V 1 •{ y = ^ (x*-- 18 x f 81) \ y^=(4 7 49 49 (x-9/7)*-t y =_4. 49 Therefore the section is a circle with radius 2/7 and center at ( 9/7, 0). Similarly, putting Z = l/4, we get a circle with radius4/3 and center at (5/3, O). So also with X = 3/4, the section is acircle, with center at radius = 148. For the general case then, i>^ Z = k,X- 4 2kx f k H ky^- x*-i 2x -1 - y^= 0 (k-l)X^ 2(k^l)x \- (k-l)y% (k-1) = d . i. X - l^xS 2 |l| X ^ i y^ ^k^j^ ^1 (4) Tx ^ k^ll^ V_ i^Zf X y*-= 4k , Therefore the sections are circles withcenters at (- 0) and r^dii 2^ Considering the projection on the xy plane, of all circles onthe surface, we have a pencil with the imaginary base points,(^,i)and (0, -i). For The intersection of {x^l)^\ y*^ 0and y^ 0 are (0,i)(0,-i). xN j\ 2x ^1 x^i y^- 2x 4 1 = V 4x = 0. This is shown hy figure 10« It follows, however, directly fromthe transformation W = z-l . For, \z-l\ = \w\ = ^(zT = constant,z moves on a circle k of the pencil with 41 and -1 as limiting pomtB. For arg.(z-l) - arg{y.\l) = const, z moves on a circle of the conju-^^gate pencil as is shown in Figure 11. Tahle of values for different values of Z = k, for radius and dist-ance of center of circle at distance k fror xy plane. K ^ l$P9 center -k-1k-1 3. 4. 21. 3. 1 .CC radii .08^ ,
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