938 
PROFESSOR A. CAYLEY ON THE SINGLE 
73. The expansions might be carried further; we have for instance 
d 0 (w) = l-|-2Q 4 cos 2S 4 cos ttv, 
d, = 1-2Q 4 , 
c 0 = 1 + 2Q 4 +2S 4 , 
13 
, + 2S 4 „ , c± =1—2Q 4 -j-2S 4 , 
, -2S 4 „ , c 8 = 1 + 2Q 4 -2S 4 , 
, —2S 4 „ , c ia = 1 — 2Q 4 — 2S 4 , 
= 2Q cos Dru-{-2Q 9 cos §7ru+2A cos \Triu-\-2v) + 2A' cos \tt(u — 2v), 
Cj =2Q+2Q 9 +2A+2A', 
= —2Q sin -|7rt£-|-2Q 9 sin fira— 2A sin \-2v )— 2A' sin ^7r(?t — 2y) 
= 2Q cos ■|7t?(+2Q 9 cos — cos -|7r(u-f-2y) — 2A' cos ^7t(u — 2v), 
c 9 = 2 Q + 2Q 9 — 2 A — 2 A', 
= —-2Q sin Dtm + 2Q 9 sin §7 tu-|- 2A sin \tt{u +2y) + 2A' sin 2v), 
$ s = 1 + 2Q 4 
d 13 =1 — 2Q 4 
do 
in which last formulae 
A=QR 4 S 4 , =^; A'=QR- 4 S 4 , = AAs ' 
Q 
Q 
74. In the single-ancl-double-letter notation we have six letters A, B, C, D, E, F; 
and the duads AB, AC, . . . DE are used as abbreviations for the double triads 
ABF, CDE, &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, 
corresponding to the double-theta functions, as already mentioned in the order 
& 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 
BD, CE, CD, BE, AC, C, AB, B, BC, DE, F, A, AD, 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, cl, e,f and two variable quantities (x, y) it is 
allowable to express the 16 functions as proportional to given functions of these quan¬ 
tities (a, b, c, d, e,f\ 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, cle 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, cl—x, e — x,f —£e=a, b, c, d, e, f ; x — y—d\ 
a—y, b—y, c—y, d—y, e—y,f—y— a /} b, c, d„ e, f,; 
