of Elliptic Functions. 261 
This last must vanish when w= a, 3w, 5w, &c., because 
then Mi = H, 3H, &c. The only factor in the numerator 
of the second member which can vanish iscanw; but this 
vanishes only for the first of the forms (A.), which therefore is 
the only admissible form. Moreover, the second of (A.) makes 
saH,sa3H, &c. = 0, which is absurd. The third of (A.) 
gives a factor infinite in both sa H, ca H, &c. 
If we develope (7.), it should give 
xv a? a 
aan ee | ¢ fi a) G im x2) a (Qa is os) . (8) 
M (l—#? 2? s*a2 4) (1—k?a*s*a4u)...(1— k2*s*a(n—1)w) 
The first formula in page 39, which is the development 
of (6.), should reduce to (8.), which vanishes for all the odd 
multiples of w. But these reductions cannot be effected, 
except for the first of the forms (A.). If we deduce (8.) im- 
mediately from (4.), as Sir James Ivory has done, we find that 
rian aaletin ant 
V1 —x* results from er 1— peo which indeed is evident. 
Sanw 
This excludes all the values of w but the first, for this factor 
must vanish when gz =sau=-+1. And hence it is evident 
that the form of M. Jacobi’s theory excludes all other values 
of w but the first. 
From what has been done, it is abundantly plain that M. 
Jacobi’s transformation is true only for the form 
ee Bate Real Wa ak 
ros tee tenis" Cae ae, > 
1 
Ci 
(5.) instead of his value, it would fail for all the other forms of 
». ‘The only possible value of w, therefore, is the preceding; 
consequently many of the forms suggested at pages 49 and 
a ; ee yes 
50 would fail, as 7K’ K+7K' K + 37K), rH tesa 
n n n n 
3K +2K! (n—1)K +2K'! 
Sia econ SO a RS 
cular transformation given in pages 52, 53, 54, 55 must fail; 
mK! v¥ —1 
and as we cannot make w = —=—— we cannot obtain a 
real transformation by means of imaginary quantities. Indeed 
(2r+1)K 
n 
would be of any utility; and this last, it is easy to see, will 
and, moreover, that if we substitute for — the general value 
, and some others. The parti- 
it does not appear that any other form than » = 
