27 
638] ON A ^-FORMULA LEADING TO AN EXPRESSION FOR E x . 
or, collecting and arranging, 
2 q 3 q 2 
4t(f 
oq* 
1 + q 1 + q” 1 +q j 1 + g 4 1 +q 5 
+ 
1-q 
+ 
J£_ + ±t + =0 
1 ~q3 T ^ u > 
an identity which it is easy to verify to any number of terms. But to prove it 
directly, we have only to add the pairs of terms in the alternate columns; calling the 
left-hand side Fq, we thus obtain 
Fq = 2q\- 
2 q 2 
+ 
3 q 4 
1 + q 2 1 4- q 4 1 + (f 
3 q i 
+ 
1 — q 2 1 — q s 
viz. this equation is Fq = 2qF(q 2 ); and thence 
Fq = 2y+ 2 F (q 4 ) = 2 y+ 2 + 4 F(q s ) = &c.; 
we thus have Fq = 0. 
The equation (B), or, what is the same thing, the equation (A) is thus proved. 
Reverting to the equation (A), we have 
4//i 
(1 + 2g + 2q 4 + ...) 4 = , 
7r 
(Jacobi, Fund. Nova, p. 188, Ges. Werke, t. i., p. 239), 
l_ 2q 2 ^ 
1 - f + 1 - q* ' 
ttK' 
(ib., p. 135; ib., p. 189), 
if q = e K , and K, E x are the complete functions FJc, EJc. 
The left-hand side of the equation is thus 
4ÜT 2 8K 2 / E x 
7T z 7T” 
_ El\ 
K)> ~ 
4K 2 
7T 2 
1 + 
2E X 
K 
)■ 
and we have 
/_ j 22^ \ _ 7T 2 1 — 9^ x — 25g 3 + 49g 6 + 81g 10 — ... 
V K ) 4 K 2 ’ 1 — q 1 — q 3 + q s + g 10 — ... 
which is a new expression for E x as a ^-function. The expression on the right-hand 
side presents itself, Clebsch, Theorie der Elasticitat (Leipzig, 1862), p. 162, and must 
f %E \ 
have been obtained by him as a value for f — IJ-jr 1 ]; th ere * s no statement 
that this is so, nor anything to show how this form of ^-function was arrived at. 
Mr Todhunter called my attention to the passage in Clebsch. 
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