73. The expansions might be carried further; we have for instance 
S- 0 (u) = 1 + 2Q 4 cos 7tu + 2S i cos 7tv, c 0 = 1 + 2Q 4 4- 2$ 4 , 
= 2Q cos \ttu + 2Q 9 cos \ttu + 2A cos | ir (u + 2v) + 2A' cos 17t (u — 2v), 
Cl =2Q + 2Q 9 + 2A+ 2 A', 
= —2Q sin \ntu + 2Q 9 sin \ttu — 2A sin \nr (u + 2v) — 2A' sin r (u — 2v), 
— 2Q cos tu + 2Q 9 cos \ttu — 2A cos ^7r (u + 2v) — 2^1' cos (u — 2v), 
c 9 =2 Q + 2 Q 9 — 2A — 2 A', 
= —2Q sin \iru + 2Q 9 sin f 7tu + 2A sin ^7r (u + 2v) + 2A' sin ^-7r (u — 2v), 
in which last formulae 
\\S- A '*& 
A = QR*S\ = ~; A' = QRSS 4 , = 
74. In the single-and-double-letter notation we have six letters A, B, C, D, E, F; 
and the duads AB, AG, ..., BE are used as abbreviations for the double triads ABF, 
GBE, &c., the letter F always accompanying the expressed duad; there are thus in 
all six single-letter symbols, and 10 double-letter symbols, in all 16 symbols, cor 
responding to the double-theta functions, as already mentioned in the order 
S- 0 1 2 3 4 5 6 7 8 9 10 11 12 18 14 15 
BD, CE, CD, BE, AC, C, AB, B, BC, BE, F, A, AB, D, E, AE, 
where observe that the single letters C, B, F, A, D, E correspond to the odd functions 
5, 7, 10, 11, 13, 14 respectively. 
75. The ground of the notation is as follows:— 
The algebraical relations between the double theta-functions are such that, intro 
ducing six constant quantities a, b, c, d, e, f and two variable quantities x, y, it 
is allowable to express the 16 functions as proportional to given functions of these 
quantities (a, b, c, d, e, /; x, y); viz. there are six functions the notations of which 
depend on the single letters a, b, c, d, e, f respectively, and 10 functions the notations 
of which depend on the pairs ab, ac, ad, ae, be, bd, be, cd, ce, de respectively: each of 
the 16 functions is, in fact, proportional to the product of two factors, viz. a constant 
factor depending only on (a, b, c, d, e, f), and a variable factor containing also x 
and y. Attending in the first instance to the variable factors, and writing for 
shortness 
a — x, b — x, c — x, d — x, e — x, f— x = &, b, c, d, e, f; x — y — Q \ 
a-y, b-y, c-y, d-y, e-y, f-y = &„ b /} c„ d /; e /; f,;