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XLVI. On the Cubic Transformation of an Elliptic Function. 

 By Arthur Cayley, Esq.* 



LET 

 _ («', b', e', d'Xx, If 

 (a, b, c, ayx, 1)^ 



be any cubic fraction whatever of x, then it is always possible to 

 find quartic functions of z, x respectively, such that 



«fe dx 



^(a,b,c,d,e][5,l)4 ~ ^(A, B, C, D, EJ^, 1)4' 



This depends upon the following theorem, viz. putting for short- 

 ness, 



U = (fl, b, c, djx, yf, 



V<={a<,b<,(^,d'X^,yf, 



and representing by the notation 



disct. (aU'-«'U, 6U'-A'U, cU'-e'U, </U'-rf'U) ; 

 or more shortly by 



disct. (flU'-fl'U, ...), 



the discriminant in regard to the facients (X, /u.) of the cubic 

 function 



(aU'-a'U, bV'-b'\], cU'-c'U, dV - d'\]J\, ixf ■ 



or what is the same thing, the cubic function 



[a, b, c, dy\, fjbf . {a', b', c', d'Jx, y)^ 



— («', b', c', d'yx, fjbf . (a, b, c, d'J^x, y^; 



and by J(U, U') the functional determinant or Jacobian of the 

 two cubics U, U', the theorem is that the discriminant contains 

 as a factor the square of the Jacobian, or that M'e have 



disct. (aU'-a'U, . .) = {J(U, U')}2(A, B, C, D, ^Jx, y)\ 



For assuming this to be the case, then disregarding a mere nu- 

 merical factor, we have 



U</U'-U'rfU = J(U, \}'){ydx-xdy), 



and the two equations give 



U</U'-U'rfU ydx-xdy 



i/di8Ct.{flU'-a'U,..)~ v/(A;B7c7n^E][a;, y)-*' 



* Communicated by the Author. 

 2 W'X 



