697] 
ON THE DOUBLE ^-FUNCTIONS. 
423 
We have then for each of the functions 
ja-z lb -z /e-z 
V d-z’ V d-z’ V d-z 
a set of four equivalent expressions, the whole system being 
Ia — z _ Va — b. a — c {V adb^ + V a^bc} _ Va — b. a — c{x — y) 
V d — z 
{be, ad) 
VadbjCj — Va^jbc 
_ Vq — 6. q - c {V abcjdj + V ad^cd} _ Vq — 6 . q — c {V acbA + V a 1 c 1 bd} < 
(a — c) Vbdbjdj — (6 — d) VacajCj {a - b) Vede^ — (c — d) Vaba^j 
/b^z 'J °a^d, “ C>) ^ bdbldl + (6 - d) V aca^j} K abc i d i “ ^ a i b i cd l 
* ^ — ^ (6c, ad) VadbjCj — Va^bc 
\J^~d ’ a ^) \/{( a —d) ^bcbiCi + (6 - c) Vadajdi} 
(a — c) Vbdbjdj — (6 — d) VacaA (a — 6) VcdcA — (c — cZ) Vabajbj 
^| {(a - 6) Vcdcjdj + (c — d) Vaba^} ^-J {Vacb^ — Va^bd} 
v d — z {be, ad) 
\/ ——^ {(q — d) VbcbjCj — (b — c) Vada^} 
V Ct — Qj 
V adbiCj — V ajdjbc 
i/S« ac) 
(q — c) Vbdbjdj — (6 — d) VacaA (q — b) Vcdcjdj — {c — d)\labajbj 
The expressions in the like fourfold form for the functions sn {u + v), cn {u + v), dn {u + v) 
are given p. 63 of my Treatise on Elliptic Functions. 
It is easy to verify first that the four expressions for the same function of z are 
identical, and next that the expressions for the three several functions 
ja — z lb — z J <p—z 
V d-'z’ V d-z' 
are consistent with each other. For instance, comparing the first and second expressions 
of JTJL. the equation to be verified is 
adb^ — aadjbc = {x — y) (be, ad), 
which is at once shown to be true. Again comparing the first and second expressions 
for \J^— we ought to have 
{(q - c) Vbdbyh + {b-d) Vaca x c a } {VadbA - Vaybbc} = {be, ad) {Vabcyb - Va^cd}.