436 
PROFESSOR CAYLEY ON THE TRANSFORMATION 
51. We may herein illustrate the following theorem, viz. we may simultaneously change 
v, a:/3:y:S:s:£ into ^ : e : h : y: & : a. 
M’ 
Thus in the equation 
— 1^, making the change, we have 
V 4 1 
that is, i m -1 )’ 
which is right. 
So in the equation ^=1 ^ , if for a moment (II), (A) are what II, A become, the 
. .8 , (H) ,, , . in' (U) 1 (A) , • , (A) 1 
equation is that is, - 9 ^-=^, or (II)=- 8 ^-IT; but obviously 
and the equation thus is (11)= — II', or say w 1 V(II)= — II' ; that is 
, „ 4 r » 12 i i 
— n' = u 
tl _L I/i r \ 
M 2 t 3 J y v s J 
+ *.l _L . 1 
' w 4 M w 7 w 7 4 i ) 4 y u s J vPtft'u* 
The general theory by q-transcendents . — Articles Nos. 52 to 71. 
52. I recur to the formula 
1 — y 1— x to. — (Zx + yx-. . +!rni (ra-1) \ 2 
L+y 1 + a; ya + (ix + ytt 2 . . + <r»R“ - *> J ’ 
and seek to express the ratios a : 0 . . . : a in terms of Writing with Jacobi a — ? 
we have in general 
a+&x + >yx\ .. +^ K ” _1) =«(l + i^) (i+i^) • • • (i + ^i).) 
(snc = sin co am ; viz. snc 2^=sn(K — 2^y), &c.), 
and the values of a, 0, . . . 0 which correspond to the moduli v 0 , v u ... v ni or say the 
values (a 0 , 005 •• • ^o) 5 (ai, 0» • . . ^), . . . (a,, 0„, . . . 0„), are obtained by giving to a the values 
»o 5 , ft> 2 . . . 
_2K 2K + iK' 4K + iK' *K' 
n ’ n ’ n ’ n 
