978 
PROFESSOR A. CAYLEY OR THE SINGLE 
115. It will be noticed that the pairs of theta-functions which present themselves 
in these equations are the same as in the foregoing “Table of the 120 pairs.” And 
the equations show that the four products, each of a pair of theta-functions, belonging 
to the upper half or to the lower half of any column of the table, are such that any 
three of the four products are connected by a linear equation. The coefficients of 
these linear relations are, in fact, functions such as the a' : -fi S 3 , or — S 3 , 2aS written down 
at the foot of the several systems of eight equations, and they are consequently products 
each of two zero-functions c. 
Thus (see “ The first set, 24 equations ”) we have 
(Suffixes 3.) 
Su , 
, Su 
X 
W 
. 
Su . 
, Su 
4 
8 
t 
= a 
1 
-s 
5 
9 
0 
12 
— a 
S 
1 
13 
3 
15 
= s 
a. 
2 
14 
7 
11 
= -8 
a 
| 6 
10 
(Suffixes 3.) 
Y Z 
_ 
SO 
SO 
(Suffixes 3.) 
a 
-s 
4 
8 
= a 3 —S 3 
a 
s 
0 
12 
= a~pS 3 
s 
a 
15 
3 
= 2aS 
-s 
a 
116. In the left hand four of these, omitting successively the first, second, third, 
and fourth equation, and from the remaining three eliminating the X 3 and W 3 , we 
write down, almost mechanically, 
Su . Su 
4 8 
0 12 
3 15 
7 11 
-(-20'S, — S 3 —ct 3 , or — S 3 
2«S, . —S'-{-a 3 , er-fiS 3 
or+S 3 , S~ —a 3 , . 2aS 
er-f-S 3 , S 3 -p or, — 2ocS 
and thence derive the first of the next following system of equations ; read 
~~c 3 c 15 d 4 d 8 
CoCiPl|d 8 
— C 4 C 8 d 4 d 8 
<¥h»Vl2 
C 4 Cg S Q S12 
~pC 0 Ci 2 d 0 di 3 
C 0 C 12^3^15 
~P c 4 c 8 d 3 d 15 
— C 3 C 15 d 3 di 5 
=-pc 4 Cg 
“■* yQ ii •—■ 0 5 
+c 3 Ci 5 ^n=0, 
= 0 , 
where the theta-functions have the arguments u, v. 
Observe that on writing herein u~0, v=0, the first three equations become each of 
them identically 0 = 0 ; the fourth equation becomes ~-c 4 2 c 8 3 -Pc 0 3 c 12 3 ™c 3 3 c 15 3 =0, which 
is one of the relations between the c’s, and which serves as a verification. 
But in the right hand system, on writing u—v=Q, each of the four equations 
becomes identically 0 = 0. 
