50 REPORT—1846. 
assumed value of y with that of the conditions required, in order that the 
quantity under the radical in the transformed expression may be equal to 
the square of an integral function of x multiplied by four unequal linear 
factors. It is shown that the number of disposable quantities exceeds by 
three that of the required conditions. But, as Poisson has remarked in the 
report already mentioned (Mem. de l'Institut. x. p. 87), and as M. Jacobi 
himself intimates, this does nut amount to an absolute @ priori proof of the 
possibility of the transformation; xo constat but that some of these condi- 
tions may be incompatible. 
Granting however the possibility of putting the quantity under the radical 
in the required form, it is shown, as in Schumacher’s Journal, that this 
condition is not only necessary but also sufficient, or, in other words, that it 
involves the second condition already mentioned. 
dy 
V(1—y*) (1 —a*y*) 
suming 7 = a U being composed wholly of odd powers of 2, and V of even 
The transcendent may be transformed by as- 
powers of it. Ifthe degree of U be greater than that of V, the transforma- 
tion is said to be of an odd order, and of an ever order in the contrary 
case. 
This being premised, M. Jacobi discusses the particular cases of the trans- 
formations of the third and of the fifth order. The first is the same as that 
of Legendre. It is shown that if we put 
_ (w+ 2u)ve + uo x 
I~ oy 8 u(v + 2u5) 2” 
where w and v are constants connected by the following equation— 
ut— ot + Quv{1—wv'} =0, 
we shall get 
‘ _ dy IL ee se 
Va-=y)d—-xy) & “v¥a—#)1—Bey 
in which k = x and A=v*. The equation connecting wu and v is called the 
modular equation. 
The “ principle of double substitution” may be illustrated by writing ai 
for x in the expression for y,; which then becomes, according to the principle 
in question, ot 
If we seek to show that the assigned value of y actually satisfies the dif- 
ferential equation just stated, we begin by finding the value of l=y. Re- 
ducing this value by means of the equation between and v, we can put it in 
the form (1 — z) a R being an integral function of x and V, as heretofore 
the denominator of the expression for y. The value of 1 + y is hence got 
by changing the sign of z, while that of 1 — vty is obtained by simultane- 
ously replacing z and y respectively by Be and a and reducing. Simi- 
uta . 
larly for 1 + v+y. Hence it will appear that 
(—-¥)U-e8y)=0-a) 1 we)... @) 
