930 
PROFESSOR A. CAYLEY ON THE SINGLE 
to + n.— = (to, n ), 
7 Tl 
to- f-l-f- n ), 
7 Tl ' 
m +(w+ 1 )-=(to, n), 
7TI 
then the formulae are 
1 + (n + l)—;= (to, n), 
'tti ' ' 
A«=A0. nn 
Bm=Bo. ii n 
Cm=Co. nn 
Dw=.D'o.Mnn 
(in, n)\ ’ 
u 
1-f 
(m, n )J ’ 
u 
1 + 
(m, n)\ ’ 
u 
(m, 7?.)J * 
where in all the formuhe m, n denote even integers having all values whatever, zero 
included, from —oo to + oo ; only in the formula for 1) u, the term for which to and 
n are simultaneously =0, is to be omitted. 
59. But a further definition in regard to the limits is required : first, we assume that 
to has the corresponding positive and negative values, and similarly that n has corre¬ 
sponding positive and negative values* ; or say, in the four formulae respectively, w T e 
consider to, n as extending 
to from —/a to /x+2, n from — v to 
33 r> P' 33 P'""l - 2, 33 53 ~~V ,, V , 
3 3 33 ”~P* 33 P 3 33 33 ^ ,, V ~j“ 2 , 
3 3 3 3 P 3 3 P 3 33 3 3 ” ^ 3 3 ^3 
where /x and c are each of them ultimately infinite. But, secondly, it is necessary 
that fi should be indefinitely larger than v, or say that ultimately -=0. 
P 
60. In fact, transforming the q-series into products as in the ‘ Fundamenta Nova/ 
and neglecting for the moment mere constant factors, we have 
* This is more than is necessary; it would be enough if the ultimate values of m were —/<, //, fi and p! 
being in a ratio of equality; and the like as regards n. But it is convenient that the numbers should be 
absolutely equal. 
