476 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704 
and the functions be accordingly called ^ 0 u, %u, %u, %u; but that, instead of this, 
I prefer to use throughout the before-mentioned functional symbols 
A, B, C, D. 
As regards the double functions, I do, however, denote the characteristics 
00 
10 
01 
11 
00 
10 
01 
11 ! 
00 
10 
01 
11 
00 
10 
01 
11 
00’ 
00’ 
00’ 
00 
10’ 
10’ 
10’ 
10 
01 ’ 
01 ’ 
01’ 
01 
11’ 
11’ 
11 ’ 
11 
by a series of current 
numbers 
0, 
1, 
2, 
3, 
4, 
5, 
6, 
7, 
8, 
9, 
10, 
11, 
12, 
13, 
14, 
15: 
and write the functions as S- 0 , A 1S accordingly; and I use also, as and when it 
is convenient, the foregoing single and double letter notation A, AB,..., which 
correspond to them in the order 
BD, CE, CD, BE, AG, C, AB, B, BC, DE, F, A, AD, D, E, AE. 
Moreover, I write down for the most part a single argument only: thus, A (u + u') 
stands for A (u + u', v + v'), A (0) for A (0, 0): and so in other cases. 
SECOND PART.—THE SINGLE THETA-FUNCTIONS. 
Notation, Ac. 
28. Writing exp. a = q, and converting the exponentials into circular functions, 
we have, directly from the definition, 
Sr q (u) = u = Au =1 + 2q cos mi + 2q* cos 27tu + 2q 9 cos Sttu + ..., 
S q (u) = S+i = Bu = 2cos tu + 2q ( cos f 7tu + 2</ 1 * * cos f 7tu +..., 
S ^ (u) = = Cu = 1 — 2q cos mi + 2q 4 cos 27tu — 2q 9 cos Smi + ...(=© {Ku), Jacobi), 
S ^ (u) = S 3 w = Du = — 2q* sin \mi + 2q f sin \mi — 2q r cos \mi +...(= — H {Ku), Jacobi), 
where a is of the form a = — a + (Hi, a being non-evanescent and positive: hence 
q = exp. (— a + /3i) = e~ a (cos /3 + i sin /3), where e~ a , the modulus of q, is positive and 
less than 1 ; cos /3 may be either positive or negative, and qi is written to denote 
exp. | (— a + (3i), viz. this is =e~* a {cos 1/3 + i sin ¿/8}. But usually /3 = 0, viz. q is a 
real positive quantity less than 1, and qi denotes the real fourth root of q. 
I have given above the three notations but, as already mentioned, I propose to 
employ for the four functions the notation Au, Bu, Gu, Du: it will be observed that 
Du is an odd function, but that Au, Bu, Gu are even functions, of u.