AND DOUBLE THETA-FUNCTIONS. 
911 
and the functions be accordingly called S 0 u, B-qa, 3-qu, B ? 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 1L 
00’ 
oo’ 
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-, B 1 . . . J 15 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, AC, 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-EUNCTIONS. 
Notation, &:c. 
28. Writing exp. a—q, and converting the exponentials into circular functions, we 
have directly from the definition 
B^(u)=Bu= Au= 1 -\-2q cos cos 27ra+2^ 9 cos 37m+ . . , 
B^(u)=B 1 u=Bu= 2q k cos ^iru-\-2q^ cos -§7m+2gv cos \ttu-\- 
^^ , ('u)=r9- 3 w=Cit=l — 2q cos Tru-\-2q^ cos 2ttu— 2(f cos . . ( = ®(Ku), Jacobi), 
^ i («)=^ 3 %=Dm= — 2g' 4 sin-^7rw+2g fl sinf7ru— 2(p cosf7ra+ . . (= —H (Km), Jacobi), 
where a is of the form a=— «+/3i, a being non-evanescent and positive: hence 
q=z exp. ( — a + /3i) = e -a (cos sin f3), where e~ a , the modulus of q is positive and 
less than 1; cos /3 may be either positive or negative, and q* is written to denote 
