594 
dx dii 
ON THE GENERAL DIFFERENTIAL EQUATION + jy'- 
[626 
where G is the arbitrary constant. It is to be observed that, in the particular case 
where e = 0, the first equation becomes 
M 2 = c + d (x -f y + z) ; 
and the two results for this case agree on putting C = c + clz. 
But it is required to identify the two solutions in the general case where e is 
not =0. I remark that I have, in my Treatise on Elliptic Functions, Chap, xiv., further 
developed the theory of Euler’s solution, and have shown that, regarding G as variable, 
and writing 
(£ = ad 2 4- b 2 e — 2bed -4- C [— 4>ae + bd +(G — c) 2 ], 
then the given equation between the variables x, y, G corresponds to the differential 
equation 
dx dy dC 
_i _i = 0 
a result which will be useful for effecting the identification. The Abelian solution 
may be written 
6 '{~ X de(x-y) F) -^ 2 + T + y)^-c~ d(x + y) = z [d + 2e{x + y) - 2Mde]; 
and substituting for M its value, and multiplying by (x — y) 2 , the equation becomes 
2de(x- y) (x dX - y dY) -e(x 2 + y 2 ) (x - y) 2 + (dX - d Y) 2 
- 2 (x 2 - y 2 ) (dX - dY) de - c (x - y) 2 - d(x + y)(x - y) 2 
= z{x-y){d{x-y) + 2e (x 2 -y 2 )- 2 (dX - dY) de). 
On the left-hand side, the rational part is 
X + Y+ c (— x 2 + 2xy — y 2 ) + d (— a? + x 2 y + xy 2 — y 3 ) + e (— x 4 + 2x?y — 2x 2 y 2 + 2xy s — y l ), 
which, substituting therein for X, Y their values, becomes 
= 2a+ b (x + y) + c . 2xy + d xy (x + y) + e . 2xy (x 2 — xy + y 2 ) ; 
and the irrational part is at once found to be 
= 2 de(x — y) (x dY — y dX) — 2 dXY. 
The equation thus is 
2a + b(x+y) + c . 2 xy + dxy (x + y) + e . 2 xy (x 2 — xy + y 2 ) 
+ 2 de(x-y)(xdY - y dX) - 2 \/XY 
z — 
(x-y) [d(x-y) + 2e (x 2 - y 2 ) - 2 (dX -dY) de] 
which equation is thus a form of the general integral of ^ ^ = 0, and also a 
particular integral of ~ = 0. 
V * v" 
dx 
dX " dY'