142 NOTE ON THE THEORY OE ELLIPTIC INTEGRALS. [657 
Thus if the second transformation be the identity z = y, w = v, M=l: we have 
v = 6u; and the equations are 
y = (x, u, 6u), Q (u, 6u) = 0, Ndy _ . dx 
V1 — y-. 1 — u 8 y 2 v 1 — x 2 . 1 — u 8 x 2 ' 
In particular, if the relation between y, x be given by the cubic transformation 
y = 
v 4- 2 u s it 6 „ 
——— Æ + — Æ 3 
v v 2 
1 4- vv? (v + 2it?) x 2 
so that the modular equation Q (u, v) = 0 is u* — v* + 2uv(l — u 2 v-) = 0 ; then, writing 
herein, v = Ou, and taking 6 a prime eighth root of unity, that is, a root of f? 4 +1 = 0, 
we have 
Q (u, 6u) = — 26 s u 2 (6ur + 6- + u 4 ); 
viz. disregarding the factor u 2 , the equation for u is u 4 4- Ou 2 4- 0 2 = 0 ; or, if w be an 
imaginary cube root of unity (or 4- <w + 1 = 0), this is (u 2 - wd) (u 2 - w 2 6) = 0 ; so that a 
value of if 2 is u 2 = — w6. 
Assuming then 6 4 4-1 = 0, v = 0u and u 2 = — w6, we have (v 4- 2u s ) v = 6 3 o) (1 4- 2w), 
v "I” 2%^ /^6 
= O'-’w (co — co 2 ); —-— = to — to 2 ; — = ® 2 , (v + 2ii 3 ) wt 2 = — to 2 (co — o> 2 ), u 8 = (o 4 6 4 — — w \ and 
the formula becomes 
giving 
_ (« — co 2 ) x 4- w 2 ox 
^ 1 — û) 2 (o) — CO 2 ) i£ 2 ’ 
dy _ (co — co 2 ) dx 
Vl - i/ 2 .1 4 <oy 2 Vl - a; 2 . 1 4- <o« 2 ’ 
where as before co 2 4- co 4-1 = 0, a result which can be at once verified. We have 
(co — co 2 ) 2 = — 3 ; or the coefficient co —co 2 in the differential equation is = V — 3, which 
is of the form mentioned in the general theorem. 
We might, instead of z = y, have assumed between y and 0 the relation cor 
responding to any other of the six linear transformations of an elliptic integral, and 
thus have obtained in each case, for a properly determined value of the modulus, a 
cubic transformation to the same modulus. 
Cambridge, 10 April, 1877.