982 
PROFESSOR A. CAYLEY ON THE SINGLE 
and this ought therefore to be the value of the first two terms, that is of 
(2Q + 2Q 9 —2A —2A')(l—2Q 4 —2S 4 ){2Q cos -g77?/ + 2Q 9 cosf ttu 
-f 2A cos j7r(« + 2'r) + 2A / cos ^rr(u— 2v)}( 1— 2Q 4 cos 77it+2S 4 cos w) 
— (2Q+2Q 9 +2A+2A')(1 —2Q 4 -|-2S 4 ){2Q cos \ttu-\-2Q? cos %ttu 
— 2A cos — 2A' cos — 2 , y)}(l — 2Q 4 cos ttu — 2S 4 cos 7 tv), 
which to the proper degree of approximation is 
= (2Q — 4Q iJ — 4QS 4 +-2Q 9 —2A—2A'){2Q cos \ttu — 4Q 5 cos \ttu cos ttu 
+ 4QS 4 cos \ttu cos 7rt’+2Q 9 cos -§7rw + 2A cos \Tr{u-\-2v)-\-2k_' cos ^ 7 t{u — 2v)} 
— (2Q —4Q 5 +4QS 4 +2Q 9 + 2A + 2A / ){2Q cos \ttu — 4Q 5 cos \ttu cos ttu 
— 4QS 4 cos \ttu cos 7rr+2Q 9 cos -§77Z( —2A cos ^7r(z/ + 2b) — 2A' cos \tt{u — 2v)} . 
This is 
(2M 0 -2X2 0 )(2M+2O) 
— ( 2 M () +2 n,,) ( 2 M ■— 2 n), =8 (M 0 H - M n 0 ) 
if for a moment 
M = Q COS ^7714 — 2Q 5 COS \ttU COS 7714+ Q 9 COS §7714, M q = Q —2Q 5 + Q 9 , 
0 = 2QS 4 cos7 ni cos 7rr+A cos § 77(44 + 2 ?;) + A' cos § 77(44 — 2v), fl 0 =2QS 4 +-A+A', 
or substituting and reducing, the value of 8(M 0 O —Mf2 0 ) to the proper degree of 
approximation is found to he 
= — 8Q(2QS 4 + A+A') cos §7744 
+ 8(Q 2 S 4 +8QA) cos §77(?/ + 24;) + 8(Q 2 S 1 '+8QA') cos \tt{ii—2v), 
which in virtue of the relations QA = A 2 S 3 , QA'=A' 2 S 2 , Q 2 S 2 = AA', is equal to the 
foregoing value of c 3 c 6 d 14 d u . I have thought it worth while to give this somewhat 
elaborate verification. 
Resume of the foregoing results. 
120. In what precedes we have all the quadric relations between the 16 double 
theta-functions : or say we have the linear relations between squares (squared func¬ 
tions) and the linear relations between pairs (products of two functions): the number 
of the asyzygetic linear relations between squares is obviously =12; and that of the 
asyzygetic linear relations between pairs is =60 (since each of the 30 tetrads of pairs 
gives two asyzygetic relations): there are thus in all 12 + 60, =72 asyzygetic linear 
relations. But these constitute only a 13-fold relation between the functions, viz., 
