208 
A MEMO IE ON THE ABELIAN AND THETA FUNCTIONS. 
[825 
where the summation extends to the four terms obtained by the cyclical interchanges 
of the letters a, b, c, d: and the like identity for ax* + fix,: 1 + yx 6 + S: there will be 
terms in aa 3 + (3d- 4- ya + S, ab s + (3b 2 + 7b + 8, ac 3 + /3c 2 + 7c + 8, but the term in 
ad s + (3d 2 + yd + S will disappear of itself. After some easy reductions, the result is 
Va DE„A K B„G n 
aa s + (3a 2 + y a + 8 
b — a . c — a 
b\c\ r , 
where the summation extends to the three terms obtained by the cyclical interchanges 
of the letters a, b, c. We have aa 3 + (3a 2 + ya + 8 = V/r. A 12 A M A 56 , and similarly for the 
other two terms; the whole equation thus divides by A m B 3S G 56 , and we find 
- A DEm = - I T ( it 
V A b —c.c—a. a —b \y A 
2 (6 - C. A 12 A 3i B 56 G K ), 
LL *\/ LL VIL 
in which equation, if we imagine ——A^, —j= B m> -~G m , each replaced by its value in 
V A V A V A 
terms of the single and double-letter functions of x x , x 2 and x 3 , x 4 , we have an equation 
of the form 
V/u. 
— _ (x 1 — X-i . X x — X 4 
Va 
x. 2 — x 3 . Xo — x 4 ) l)E m — 
1 
rp rp rp rp rp rp rp . rp 
Wf tA/g • lA/J tDq . lA^2 tA/g • tA/g l 
M, 
where I is a given rational and integral function of the single and double-letter 
functions of x x , Xo. and x 3 , x 4 . The factor on the left-hand side has been made the 
same as in the formula for the single-letter functions A m , &c., and to do this it was 
necessary to bring in on the right-hand side the factor 
1 
X x — X 3 . X x — Xi . X 2 — X 3 . Xo — x 4 ’ 
this disappears in the expression for the ratio of two double-letter functions; but it 
enters into the expression for the ratio of a single-letter to a double-letter function, 
and it then requires to be itself expressed in terms of the functions of x ly x 2 and 
x 3 , x 4 : it is easy to see that we have 
rp rp rp rp rp rp rp rp 
tL- I tA/g • t4/^ • tA/g *A/g • * /t/ 2 *A/4 
2 (A 2 M B\ 2 - A\o_B 2 34) (A\G\ 2 - A\ 2 G 2 U ) 
(a — bf (a — cf 
where the summation extends to the three terms obtained by the cyclical interchanges 
of the letters a, b, c : these being a set of any three out of the six letters. 
We have, in what precedes, obtained the expressions for the ratios of the 16 
functions Age, ..., F 56 , AB m ,DEw in terms of the ratios of the like functions of x x , x 2 
and x 3 , x A .