424 
PEOFESSOE CAYLEY ON THE TEAN SFOEMATION 
viz. we thus obtain the required value of ^ as a rational fraction, the denominator 
being the determinate function F'v, and the numerator being, as is easy to see, a deter- 
minate function of the order n as regards v. 
32. The method is applicable when M is only known by its expression in terms of q; 
but if we know for M an expression in terms of v, u, then the method transforms this 
into a standard form as above; and byway of illustration I will consider the case n= 3, 
where the modular equation is 
v i + 2v 3 u 3 —2vu—u i =0, ’ 
and where a known expression of M is . Here writing S_„ S 0 (=4), S, &c. 
to denote the sum of the powers — 1, 0, 1, &c. of the roots of the equation, we have 
S^=S 0 d-2w 3 S_„ =0 , as appears from the values presently given, 
S^=S 1 + 2m 3 S 0 , =6u 3 , 
sg=S 2 +2 W 3 S 1 , =0 , 
S^=S,+2w"S s , =6 u; 
and observing that v 0 being ultimately replaced by v, we have 
Svj=Sv 0 — v, 'wSw 0 -(-'y 2 , vSv 0 Vj + vj$v u — v 3 , 
that is 
Sv 1 =—2u 3 —v, Sv 1 v 2 =2u 3 v +w 2 , v 1 v 2 v 3 =2u—2u 3 v 3 —v 3 , 
we have 
IV M= ( S 3+2^ 3 S 2 ) 
+ (2tf+«XS>+2u 8 S 1 ) 
+(2trt>+^XS 1 + 2i^S 0 ) 
+ ( - 2 u + 2wV+O(S 0 + 2w 3 S_ 1 ), 
viz. this is 
2(2u 3 +3 ^ 3 (S 0 +2 m 3 S_ 1 ) 
+ ^ 2 (Si + 4 m 3 S 0 + 4w 6 S_ J 
+v(S 2 -f-4w 3 S l +4w 6 S 0 ) 
+ (S 3 + 4m 3 S 2 + 4w 6 S, - 2 mS 0 - 4 w 4 S_ J . 
But we have 
S_ 1 = -| 3 , S 0 =4, S 1= -2«i 3 , S 2 =4< S 3 =.e«-8w 9 ; 
and the equation thus is 
(2 u 3 + 3 — w) ^ = 3 (v 2 v ? -f 2u 5 v + 1 )u ; 
