ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 341 



To prove this, wc remember the well-known expression 



fl^(.r) is derived from 0(a-) by changing q into —q. 

 Now if we change q into —q, 



/ 2Kk' becomes \/—, F' = K becomes KA.-', 



T" 1 



E' becomes -r— *, A becomes — , 

 i:' A 



E^ = 1 c?0 V 1 — !<? sin* becomes a; I ^ > 

 Jo Jo 



and P ^ = 1 Ed,- ^ "^"'^"""'^ -K 



and therefore E^ becomes 



1 T-, P sin (A cos d) 



F^ becomes FF^, and ^ becomes A-' ^. 



A A 



Substituting all these in the expression for Qix), we have 



0, 



(P Eif (f(p _ 1 E' 



r(cp)!'+iog A 



(P 



A 2 K ''^' "« 



A 2 K ^ . 



which is, in fact, Jacobi's expression for QJ^oc). In the memoir now under 

 consideration, Jacobi arrives by partial differentiation at the following theorem: 

 Let 



^/ 



(l_<)V-^-i(l_H)~V^, 





y 



* Fundamenta Nova, p. 111. 



t Hymers's Integral Calculus, p. 220. 



