554 
ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS (SEQUEL). 
[870 
83. There exists a remarkably simple particular case. Write for convenience 
6 = \J7, the pa-equation is satisfied by the values p = — 6, a = -fd. In fact, these 
values give 8pa = 3d 2 , =21, -y = (ff — 9) 4- (ff — 1), =513; the term in a is thus 
21.513, = 10773; but, assuming p 2 = 7, we have 
p 8 _j_ i4p 6 4- 63p 4 + 70p 2 — 7 = 2401 4- 4802 + 3087 + 490 — 7, = 10773, 
and the equation is thus satisfied. And these values, p = — 0, a = — §d, give p 2 = 7, 
¡3 = §d, A x = 2d, A 2 = pd ; the equation 12A 2 = 6A.J 2 4- 8aA 1 — p 4 4- 7 thus becomes 
12pd= 168 — 42 — 49 4-7, =84; that is, pd = 7, = d 2 , or p = d(=—p). We have a 2 —1, 
— 1 > — — e 1 " 
F4 > 
but from the equation p—p 
a? — 1 
it appears that the sixth roots 
must be equal with opposite signs, say s/d 1 — 1 =k , v 7 ^ 2 — 1 
— 
T ' 
Retaining d to 
stand for its value = y7, the differential equation is 
and it is satisfied by 
dy _ 6dx 
Vi — I dy 2 + y 4 V1 + I da? 2 4- a? 4 ’ 
_ a? (d 4- 7a? 2 4- 2dar* + a? 6 ) 
y 1 + 2da? 2 + 7a? 4 + da? 6 
It may be remarked that the quartic functions of y and a? resolved into their linear 
factors are 
si + d 
and 
Si-0 If 
Si+ 6 
\ y + 2 V2 (1 + 7)1 \ y + 2 V2 (1 + 7)j j 2 ' + 2 V2 (1 -*)) f + 2 V2 (1 - 7) 
37 — d 
a? 4- 
3 - 7d 
{ 2 V2 (1 + i) 
æ+ 1 
3 — id 
2 a/2 (1 -7) 
a? + 
3 + 7d 
2 V2 (1 - 7) 
and that for the first of the y-factors, substituting for y its value, we have 
a? 7 + 2da? 5 4- 7a? 3 + da? 4- 77—ff + , , N (da? 6 4- 7a? 4 4- 2da? 2 + 1) 
2 V2 (1 4- 7) 
3 — 7d if 
1 +7d 
1 4- 71 2 
= ^ + 2V2(l + «)i r + 
with like expressions for the other y-factors respectively. 
Brioschi’s Transformation Theory. Art. No. 84. 
84. M. Brioschi has kindly referred me to two papers by him, “ Sur une Formule 
de Transformation des Fonctions Elliptiques,” Comptes Rendus, t. lxxix. (1874), pp. 
1065—1069, and ibid. t. lxxx. (1875), pp. 261—264. They relate to the form 
dx _ dy 
V4a? s - g 2 x -g 3 V4y 3 - (x 2 y - (J 3 ’