f7 04 
704] 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
491 
We 
Addition-formula}. 
56. The addition-theorem for the quotient-functions is of course given by means 
of the theorem for the elliptic functions: but more elegantly by the formulae relating 
to the form dx-i-Va — x.b — x.c — x.d — x obtained in my paper “ On the Double 
^•-Functions” (Crelle, t. lxxxvii. (1879), pp. 74—81, [697]); viz. for the differential 
equation 
dx dy dz 
—= A — = 0 
VX VF fz 
to obtain the particular integral which for y — d reduces itself to z — x, we must, 
in the formulae of the paper just referred to, interchange a and d: and writing for 
shortness a, b, c, d = a—x, b—x, c — x, d — x, and similarly a,, b,, c /} d^a — y, 
b — y, c — y, d — y, then when the interchange is made, the formulae become 
Id-, 
V ^ 
fd — b.d — c (Vadb / c / + Va/^bc} 
(6c, ad) 
s/d-b.d-c.x — y 
V adb / c / — V a^bc 
fd — 6. d-c {Vbdc/q + Vb d,ca| 
’ (d- c) Vabab - (6 - a) V cdc,d, 
\ld - 6 . d — c (Vcda,b, + Vabc^J 
(d — b) Vaca/3, — (c — a) Vbdb^d, 
lb - z 
V a — z 
\J d d~a “ c ) ^ aba A + ( 6 “«) ^cdc,d,} 
(6c, ad) 
\J fZTa K bda , c /~ Ad.ac} 
V adb / c / — V a / d / bc 
bd) 
(d — c)\! aba,b, — (6 — a) Vcdc / d / * 
“ a ) ^ bcb / c / + (6 - c) Vaba / b / } 
(^ — 6) Vaca^ - (c — a) Fbdb / d / 
■A 
'd-c 
d — a 
{(d — 6) Vca^a, + (c — a) Vbdb / d / | 
'd-c 
d — a 
(be, ad) 
{Vcda/b, — Vabc/iJ 
Vadb c, — Va/ibc 
\ ^—a ^ — a ^ ^ bcb / c > — (b — c ) Vada/iJ 
(d — c) Vabab, — (6 — a) Vede^ 
d — c 
d — a 
(a,b, cd) 
(d — 6) Vaca / c / — (c - a) Vbdb / d / 
62—2