On the Reduction of the Hyperelliptic Integrals P3 to Elliptic Integrals By T

William Gillespie

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Also r = ^=~- (J, ip^^ - T, x') (24) has for its first polar ^with regard to an arbitrary parameter the cubic involution itself, i . E F ^ F. . . '. The reducible integral § 1 (4) has the form {ip — X) {Xdx) [ I'Cs'V + '. X) r=r, r^-3r;. (25) — 12 - The transformation (26) Z/i = U y, = V can by a linear transformation of the elliptic integral, he thrown into the form r ^ r ==0, which may be expressed thus, — %Cr y ''' — * dx, y^ ~ ^ dx^' Let us examine the form of the elliptic integral. Before ...reduction we have the hyperelliptic in the form ■0- {xdx) i {xd, y {xe, ) (xd. ^^^ (xe, ) r^Tj^'r' and by the transformation (26) X e, . -. Q (xd, Y K^e, ) {xd. ^^ {xe^ = (ye\) ye, ) = {L, ip^y) + Liy)) (28) and r, ^ r^ r. - ^ j^^^' ^ ^^X) (yfi) = say (?/) and we also have (29) 0/'/^) ^ i^i> {ocdx) ^-) . '. After reduction our elliptic integral has the form ^ ijiM (30) ^ J ^~^, ipiy) + J, x(y)] g^y) where C is some constant. The expression F ^ F ' ^ FT ' can be thrown into a more serviceable ^ X X /.

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