254 
ON A DIFFERENTIAL EQUATION 
[597 
But knowing a particular value of Q we have 
z — exp 
Qdz 
a particular value of z, and thence in the ordinary manner the general value of z, 
giving the general value of Q. 
The solution given in my former paper may be exhibited in a more simple form 
by introducing, instead of k, the variable a connected with it by the equation 
a 3 (2 +a) 
k 2 — —We have in fact, Fundamenta Nova, p. 25, [Jacobi’s Ges. Werke, t. l, 
p. 76], 
u 8 = a 3 
v* = a 
2 + a 
T+2a’ 
2 + a. 
1 + 2« 
\ 
viz. these expressions of u, v in terms of the parameter a, are equivalent to, and 
replace, the modular equation u 4 — v 4 + 2uv (1 — mV) = 0. We thence obtain 
that is, 
?y 8 ?)8 = a4 ( 2 + «) 4 ^ (2 + «) 2 
(1 + 2a) 4 ’ u 8 a 2 (l + 2a) 2 ’ 
uv = \/( a ) 
V2 + a\ 
v 2 _ i / 
i 2 + + 
U + 2a) ’ 
v? V(a) V 
U + 2a) 
and the particular solution, Q = — 2 + 2uv, becomes 
'll* 
«=v+ V(1 + 2 “' 2+a) ’ 
Introducing into the differential equation a in place of k, this is found to be 
J5 + 2 
(«+1)1 
I 
V a/j 
— + a 2 + 2 (—ha 
Q*-Q 
5 + 2 f a + - 
- 3 = (1 - a 2 ) 
•¡5 + 2 
(«+1)1 
( 
V ayj 
dQ 
da ' 
But from this form it at once appears that it is convenient in place of a to introduce 
the new variable /3, = a + -; the equation thus becomes 
satisfied by Q = \J(o + 2/3); or, what is the same thing, writing 5 + 2/3 = y 2 , that is, 
/3 = — f + y 2 , the equation becomes 
4<? + |( 3 + 6 7 *-7‘)-12=-(y‘-l)(7’-9)^. 
satisfied by Q = y.