46 
[808 
808. 
NOTE ON A FORM OF THE MODULAR EQUATION IN THE 
TRANSFORMATION OF THE THIRD ORDER. 
[From the Messenger of Mathematics, vol. xii. (1883), pp. 173, 174.] 
In my Treatise on Elliptic Functions, pp. 214—216, writing only j, ~ instead of 
il, O', and a, /3 instead of a!, /3', I have shown as follows: viz. if k, \ denote as 
usual the original modulus, and the transformed modulus, and if 
T _(k A + 14& 2 + l) 3 T , (X 4 + 14X, 2 + l) 3 
J ~ T08& 2 (1 - k-y ’ _ 108X. 2 (1 — X. 2 ) 4 ’ 
then the relation between J and J' can be found by the elimination of cl, /3 from 
the equations 
a + /3= 1, 
(1 + 8a) 3 v (1+8/3) 3 
64a (1 — a) 3 ’ 64/3 (1-/3) 3 * 
By a very slight change we obtain the result given by Prof. Klein in his paper, 
“Ueber die Transformation der elliptischen Functionen, &c.,” Math. Ann. t. xiv. (1879), 
pp. Ill—172; viz., see p. 143, the relation is to be obtained by the elimination of 
t, t from the equation W = 1, and the equations 
J : J -1 : l=(r — 1)(9t -l) 3 : (27r 2 - 18r - l) 2 : -64r; 
J' : J'-l : 1 = (t — 1) (9t' — l) 3 : (27r' 2 -18r'-1) 2 : -64r; 
these last equations being equivalent to two equations only in virtue of the identity 
(t — 1) (9t — l) 3 + 64r = (27t 2 — 18t — l) 2 , 
and the like identity in r. 
In fact, writing a = —, /3 = , T n , the equation a + /3 = 1 becomes tt — 1; and 
then for a, /3 substituting their values, we have 
T- (9t— 1) 8 (t —1) v _ (9t-pHr-l) 
- 64t ’ - 64r' 
which are the formulse in question.