OF ELLIPTIC FUNCTIONS. 
445 
f a positive integer, and the summation extending as before to the n-\- 1 values of v, 
and corresponding values of -. This is a rational function of u, and it is also integral. 
As to this observe that the function, if not integral, must become infinite either for u— 0 
(this would mean that the expression contained a term or terms A u~ a ) or for some 
finite value of u. But the function can only become infinite by reason of some term or 
d 1 
terms of Sv f ~ becoming infinite ; viz. some term — ~ — must become infinite; or attend- 
« ° ’ snc 2gw 
ing to the equation 
«=w B {snc 2 a snc 4« . . , snc(ra— l)a>}, 
it can only happen if u— 0, or if «=co ; and from the modular equation it appears 
that if v= co , then also u= oo: the expression in question can therefore only become 
Q ry 
infinite if w=0, or if u= oo . Now u = 0 gives the ratios each of them a 
determinate function of n, that is finite ; and gives also t»=0, so that the expression does 
not become infinite for u— 0 ; hence it does not become infinite either for u— 0 or for 
any finite value of u ; wherefore it is integral. The like reasoning applies to the sum 
0 ... . 
Sv~ f - ; viz. this is a rational function of u ; and it is quasi-integral, viz. there are no 
a 
terms having a denominator other than a power of u, the highest denominator being 
u n/ ; viz. the expression contains negative and positive integer powers of u, the lowest 
power (highest negative power) being ~ f - 
65. It is to be observed, further, that writing the expression in the form 
(where S' refers to the values .v„ v 2 , . . . v n of the modulus), and considering the several 
quantities as expressed in terms of q, then in the sum S' every term involving a frac- 
h_ 
tional power q n acquires by the summation the coefficient (1+a+a 2 ... -}-a re-1 ), and 
therefore disappears ; there remains only the radicality (£■ occurring in the expressions of 
the v’s ; and if nf=q> (mod. 8), ^ = 0, or a positive integer less than 8, then the form of 
the expression is q* into a rational function of q. Hence this, being a rational and 
integral function of u , must be of the form 
Au 11 + 8 + 16 -f- &c. 
66. We have thus in general 
S?/^=A^+B^ +S + &c.; 
and in like manner 
Sv ~ f ~=A'u~ nf + B'u~ nf+8 + &c. 
MDCCCLXXIV. 
B N 
