OF ELLIPTIC FUNCTIONS. 
443 
We have in what precedes the complete y- transcendental solution for the transformatio 
secunda ; viz. the original modulus k(—id) being given as a function of q, then, as well 
the new modulus X n ( = v* n ) and the multiplier M,„ as also the several functions which 
enter into the formula 
are all expressed in terms of q. The expressions all contain q 7 , and by substituting for 
this an imaginary wth root of q, we have the formulae belonging to the several (n— 1) 
imaginary transformations. 
63. As an illustration of the formulae for the transformatio secunda I write n— 7 ; and 
putting for greater convenience q=r 7 , that is r=f, then we have 
*V=n/2 
J 
M 
where 
snc 2 w. 
=2/W, 

snc 4co ? 
= 2/V)B, 
snc 6u>, 
=2f 2 (r 7 )C, 
A =r 
5.19... 9.23.. 
2 . 16 . . . 12 . 26 . 
B=r 2 3 17 ■ n - 25 -- 
'4.18.. 10.24..’ 
1.15.. 13.27.. 
' 6 . 20 .. 8 . 22 . .’ 
where the numerator of A. denotes (l+r 5 )(l-f r 19 ) . . (l + r 9 )(l+r 2 - 3 ) . . , and so in other 
cases, the difference of the exponents being always =14. And we have, as mentioned, 
the identical equation 
The values of the several expressions up to r 50 are as follows. Mr. J. W. L. Glaishee 
kindly performed for me the greater part of the calculation. 
