592 
[626 
626. 
ON THE GENERAL DIFFERENTIAL EQUATION + % = 0, 
JX J 1 
WHERE X, Y ARE THE SAME QUARTIC FUNCTIONS OF 
x, y RESPECTIVELY. 
[From the Proceedings of the London Mathematical Society, vol. vm. (1876—1877), 
pp. 184—199. Read February 8, 1877.] 
Write B = a+bd+ cd 2 +dd s + ed 4 , the general quartic function of d; and let it be 
required to integrate by Abel’s theorem the differential equation 
We have 
a particular integral of 
dx 
VX 
+ d y = 0 
VP 
x 2 , x, 
1, \/x 
y 2 > y> 
l, VP 
z‘ 2 , z , 
1, nZ 
tv 2 , IV, 
1, >JW 
= 0, 
dx du dz dw 
1 i- 4- 1 = o 
VX VP V^ VTE 
and consequently the above equation, taking therein z, w as constants, is the general 
integral of 
dy_ _ 0 
vx VP 
viz. the two constants z, w must enter in such wise that the equation contains only 
a single constant; whence also, attributing to iu any special value, we have the general 
integral with z as the arbitrary constant.