446 
PROFESSOR CAYLEY ON THE TRANSFORMATION 
We may in these expressions find a limit to the number of terms, by means of the before 
mentioned theorem that we may simultaneously interchange u, v ; a, 0, . . . g, a into \ i ; 
0 
-r, g>, . . . 0, a. Starting from the expression of Si/-, let (p be the corresponding coeffi- 
cient to 6 ; viz. in the series os, 0 . . 6 . . <p . . g, <r, let <p be as removed from a as 6 is from a ; 
then the equation becomes 
Su _/ ^=Aw -,t 4-Bw _ ' x " 8 + &c., 
where -=- ~=~z - ; the equation thus is 
<r a cr u n sc u 
and by what precedes the series on the right-hand side can contain no negative power 
higher than — ; that is, the series of coefficients A, B, C . . . goes on to a certain point 
only, the subsequent coefficients all of them vanishing. 
In like manner from the equation for Sv~ f - we have 
Su /+1 1 =AW n+ » f +B'u& +i)f '« + &c., 
where the indices must be positive ; viz. the series of coefficients A', B', . . goes on to a 
certain point only, the subsequent coefficients all of them vanishing. 
67. The like theory applies to the expression i. We have, putting as before 
nf = (a (mod 8), 
S 'y / -^=AM' i +Bw ,1+8 -f-. . . , 
8v: f ft=A!u-*+B'u-* +9 +. . . 
and we find a limit to the number of terms by the consideration that we may simul- 
taneously change u, v, ~ into -, -, ; the equations thus become 
M u v w 4 M u 
^ where, if/= or <4, there must be on the right-hand side no negative power of u; but 
if f > 4, then the highest negative power must be , and 
S^ + '^=AV^+B'^ 4 +. . . , 
where on the right-hand side there must be no negative power of u. 
