442 
PROFESSOR CAYLEY ON THE TRANSFORMATION 
(1— ^ 2 ) . . is of q ; also 
n(i+ 2 ' + ¥)..(i+ 2 -?)..=^ 
(!+<?»).. 
where (1 -\-q n ) . . denotes (1 -l-g«)(l -\-q n )(l -\-q n ) • • , viz. it is the same function of q » that 
(1+g) . . is of q; and the like as to the denominator factors : we thus have 
[(.-+ 
. (l+g») . 
. (1+g 2 ).. (1-g).. i 
1(1— ff 2 )- 
. (1+g).. 
. (1 +g«) . . (1— g") . .J 
viz. this is 
or we have 
[(1-g 2 )- 
.(1+g).. 
1(1 +g 2 ). 
• (1-si)- 
l 2 
l 2 
^=tf(q n ) + f(q), 
■agreeing with a former result. 
We have 
2/3„— jyj - — 1, that is 
2 f 1 1 _L _l 1 _ ft 2 (g”) y 
1 snc 2w n ‘ snc 4co n ' ' " ~snc [n — l)co„J <p 2 (g) ’ 
a result which, substituting on the left-hand side the foregoing values of the several 
functions, must be identically true. 
62. We have also 
v n =u n { snc c la n snc 4 u n . . . snc (n— l)a n }, 
where the term in { } is 
2qn (l + g + »)..(l +q ») . . 
or observing that the sum of the exponents - is -{1 + 2 . . — , this is 
=/- +1 (z )• 
(1+g”) • -.(1+g) .. 
(V'2 J*- 1 }" (l + g 2 )-*(l + g™)-- 
_1 
or, the last factor being f(q n ) +f(q), the expression is 
/~”(g)^ 7^n-i z~* +rn f(r) > 
or, multiplying by u n , =(\/2 f q* f\q), we have 
^=V2^/(r), 
agreeing with a former result. 
