578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 119 
Equation-systems for the cases n = 3, 5, 7, 9, 11. Art. Nos. 8 to 10. 
8. u — 3, cubic transformation. h = u 4 , fl = — 2 (here and in the other cases). 
P = a, Q = /3. The condition here is 
kWar + (2a/3 + /3-) = fl/j {(« 2 + 2 a/3) + ¡3'\x 2 }, 
and the system of equations thus is 
/¿a 2 = fl/3 2 , 
2a/3 + /3 2 = fl& (a 2 + 2a ft), 
and similarly in the other cases; for these it will be enough to write down the 
equation-systems. 
n = 5, quintic transformation. 
P = a + y# 2 , Q = /3. 
Pa 2 = fly 2 , 
2ay + 2 a/3 + /3 2 = i1 (2ay + 2/3y + /3 2 ), 
y 2 + 2/3y = fl& 2 (a 2 + 2a /3). 
% = 7, septic transformation. 
P = a + y# 2 , Q = /3 + Sx 2 . 
Pol 2 = f!8 2 , 
k (2ay + 2a/3 4- /3 2 ) = fl (y 2 4 2y8 + 2/38), 
y 2 + 2/3y + 2a8 + 2/38 = fl& (2ay 4- 2/3y + 2a8 4- /3 2 ), 
8 2 4- 2yS = flP (a 2 + 2a/3). 
7i = 9, enneadic transformation. 
P = a 4- y« 2 + eP, Q = /3 + Sx 2 . 
Pa 2 = fie 2 , 
P (2ay + 2a/3 4- /3 2 ) = fl (2ye 4- 2e8 + 8 2 ), 
2ae + y 2 + 2a8 + 2y/3 4 2/38 = fl (2ae 4 y 2 4 2yS + 2e£ + 2/38), 
2ye + 2y8 + 2e/3 + 8 2 = flA; 2 (2ay + 2a8 + 2y¡3 + /3 2 ), 
e 2 + 28e = flAr* (a 2 + 2a/3). 
7i = ll, endecadic transformation. 
P = a + yx 2 + ex A , Q = /3 +Sx 2 + 
Pol 2 = fl£ 2 , 
A,- 3 (2ay + 2a/3 + /3 2 ) = fl (e 2 + 2e^+ 28£), 
A; (2ae + y 2 4- 2a8 + 2y/3 + 2/38) = fl (2ye + 2y£ + 2e8 + 2/38 + 8 2 ), 
2ye + 2a£+ 2y8 + 2e/3 + 2/3£+ 8 2 = fl/j (2ae + y 2 + 2af + 2y8 + 2e/3 + 2/38), 
e 2 + 2y£ + 2e8 + 28£= fl P (2ay + 2a8 + 2y/3 + /3 2 ), 
2e£ + ? = HP (a 2 + 2a/3). 
And so on.