j^o4ifti\noV,«ftA of an Elliptic Function, 45 



Malce «=0, 2w', 4^', &c. ; and we have7/ = 0. Again, make 

 u=oo'f 3a)', 5co', &c.; and we have 3/= + 1. For the other 

 forms of CO we have 



_ sa.usa{u + 2co) s a (w-f (2?j — 2)«;) , 



^ ~ srt(K-2co)sa(K-4«j) sa(K-(2w-2y«ry ' ^'^'^ 



When w= K, K — 2 w, &c., this gives 3/= + 1. But which 

 of these values falls between ?i = 0, u = 2 co, or between m=0, 

 w = 2 w', we cannot tell. We ought, however, in good facto- 

 rial formulae to know. And when u = oo', 3 w', &c., this must 

 give j/= + 1 ; because (3.) must give the same value of 3/ thai 

 (2.) does. But when we put for w' its value in w, this does 

 not appear; nor can we reduce the result, except for the second 

 form of cu, without eliminating w, which would be equivalent to 

 reducing (3.) to (2.) But all these things ought to be appa- 

 rent in the face of factorial formulae. Other faults might be 

 pointed out; and we might point out similar faults in the ex- 

 pression of the value of v^l —3/^. But I shall not dwell upon 

 the subject. If the last three forms of 00 do not render the 

 values of 3/ erroneous, they render the formulae faulty. They 

 are unnatural; they reduce to the first form, and there is but 

 one transformation. 



If Mr. Cayley had proved in his paper of November 1844 

 all that he wished, it would not follow that the different forms 

 of CO would give really distinct transformations. But his for- 

 mula (6.) is not Jacobi's. To agree with his, the first member 



should be d), ( -=-5= I* the denominator of the second 



*'(m) 



<^(K— 2co)$(K— 4«)) ... <^(K-(2w — 2)co). 



He might however, by a suitable modification of (5.), have 

 arrived at Jacobi's result. But his proof of the possibility of 



is not to me satisfactory. It is unnecessary to state my ob- 

 jections, because he has not shown the possibility of 



2 7w K + (2 7«'+ 1) K' \/'^ + 2 r6 = 2ftH -I- (2 j[a' + 1)H' v"^. 



The omission of this appears to me very strange. Ot course 

 the omission renders his paper perfectly nugatory. For the 

 possibility of this equation does not follow from that of the 

 former, supposing that to be fairly proved. If he had shown 

 the possibility of this last, I believe he would have found it 

 necessary to modify a little the form of some of the quantities 

 a, bf &c. Moreover, if he had transformed the factorial values 

 ofy{M), F(w) (I use here Abel's symbols), he might have 

 found it necessary to modify them stjU further. In a com- 



