A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
493 
704] 
then the formulie are 
Au = .40. 
Bu = BO . 
Cu = CO . 
Du = D'0.uUU\l + 
(m, n) j 
where in all the formulae, m and n denote even integers having all values whatever, 
zero included, from — oo to +00; only in the formula for Du, the term for which m 
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 m has the corresponding positive and negative values, and similarly that n has 
corresponding positive and negative values*; or say, in the four formulas respectively, 
we consider m, n as extending 
m from — fjb to /jl + 2, n from - v to v + 2, 
» >> /u >> + » >> ^ 
» >> № » ^ ^ d” 2, 
>> ““ № ? ?? ?? ~~~ V Vy 
where ¡i and v are each of them ultimately infinite. But, secondly, it is necessary 
that /a should be indefinitely larger than v, or say that ultimately - = 0. 
60. In fact, transforming the g-series into products as in the Fundamenta Nova, 
and neglecting for the moment mere constant factors, we have 
Au = (1 + 2q cos iru + g 2 ) (1 + 2g 3 cos iru + g 6 )..., 
Bu = cos \iru (1 + 2g 2 cos 1ru + g 4 ) (1 + 2g 4 cos iru + g 8 )..., 
Cu = (1 — 2g cos iru + g 2 ) (1 — 2g 3 cos iru + g 6 )..., 
Du = sin \itu (1 — 2g 2 cos iru + g 4 ) (1 — 2g 4 cos iru + g 8 )..., 
and writing for a moment a=—.. and therefore g^+g = e^ ai + e ’~ nal , — 2 cos ^7ra, &c., 
each of these expressions is readily converted into a singly infinite product of sines 
or cosines 
Au = II. cos \it (u + no), 
Bu = II. cos \ir {u + na), 
Cu = II. sin ^7r (u + na), 
Du = II. sin \ir (u + na), 
* This is more than is necessary; it would be enough if the ultimate values of m were — +> /* ail d + 
being in a ratio of equality; and the like as regards «. But it is convenient that the numbers should 
be absolutely equal.