364 On the Cubic Traiisformation of an Elliptic Function. 



whence writing z for U' : U, and putting y equal to unity, we 

 have 



dz dx 



i/disct.(fl^-a',..) ^(A,B,C,D, EJa?, 1)^' 

 where disct. {az — ol, . . .), or at full length, 



disct. {az—a', bz—b', cz — c', dz — d'), 

 is a given quartic function of z, 



= (a, b, c, d, ej^z, 1)"* 



■ suppose, which proves the theorem of transformation. 



The assumed subsidiary theorem may be thus proved : sup- 

 pose that the parameter 6 is determined so that the cubic 



V + dV 

 may have a square factor, the cubic may be written 

 (a + 0a', b + eb' c + dc', d+ed'Jx, yf, 

 and the requisite condition is 



disct. (a + ^fl', . . ) = 0. 

 There are consequently four roots; and calling these 6y^,6c^,6^,6^, 

 we have identically 



disct. ia+ea', . .)=\v{e-e,){e-e^){e-d^){d-e^), 



or what is the same thing, 



disct.(flU'-fl'U,..)=K(U + ^iU')lU + ^2U')(U + ^,U')(U+^4U'). 



Now any double factor of U or U' (that is the linear factor which 

 enters twice into U or U') is a simple factor of J(U, U'), and we 

 have J(U, U') = J(U, U + ^U'), and consequently 



J(U, U') = J(U, U + ^,U') = &c.; 



hence the double factors of each of the expressions U + ^jU', 

 U + ^gU', U + ^gU', U + ^JJ' are simple factors of J(U, U'), 

 or what is the same thing, J(U, U') is the product of four linear 

 factoi's, which are respectively double factors of the product 



(u + ^lU') (U + d^\]') (u + d^V) (u + ^^u'), 



or this product contains the factor {J(U, U')}^, which proves 

 the theorem. 



2 Stone Buildings, W.C, 

 March 5, 1858. 



