1198 
MR. R. F. GWYTHER ON THE DIFFERENTIAL 
differential equation of conics, Mr. Elliott does, in fact, illustrate these methods of 
procedure. 
§ 4 . AiDplication of Results to the Cubic. 
21 . The differential equation of the general cubic. 
It has been previously shovm that the order of this differential equation will be 9 , 
the degree 10, and the weight 35 . It is also a function of u^, U^, Ug, and Ug, since 
it is easily seen that u^ cannot explicitly enter the equation. 
Written as a determinant, the equation is 
CIq, 
2a3a6+2aj,a5, 
af-f^a^a.-f^a^a 
ag, 
arj, 
(Xg, 
a/+ 20^3%, 
2 a^a^ + 
a^, 
(Xg, 
Srqfq, 
af + 2a3fq, 
a^, 
« 5 > 
« 3 b 
cqng. 
« 5 > 
« 3 . 
0 , 
a ^ 
— 
0 
a^ 
3 a/cq + da.g:if 
0 
0 
= 0 . 
From this we see that «§ and ttg enter only in the form — af 
Hence we have the terms — Ug^ + ■ • • Since the degree of these terms 
is 24 , and weight 72 , the differential invariant when complete must be divisible 
by u,Hf 
The differential equation therefore takes the form 
n/U,Ug - (Ug^ + 4U/) + EF,^Ug -h = 0. 
Determining E and T, either by the condition of containing u^ as further factors 
or by comparison with the determinant, we obtain 
u-^VrfJ, - Ug^ - 4U73 - ISUgMTg - 56U58 = 0, 
or 
u/U,Ug - Vg2 - 4U,3 + u,Wg - 0.(36), 
where 
Vg = Ug + Swgh 
22. The first integrals of the differential equation of the general cubic. 
From the equation to the general cubic, found as a differential covariant, it is 
possible to find a complete set of the first integrals of the differential equation. 
For the equation to the cubic being written in the form 
^ iv - - y + -f - y f- yd^f - x) 
+ {1 — y\^ — y -f yn) + «(^ — 
+ 3Z>3 (77 - ^ - y + yfff + 6 m (v - yi i - y yi^) - x) 
+ 3^3 a — xf + Scfi7] — ijff — y-\- y^x) = 0 , 
* ‘ Messenger of Mathematics,’ vol. 19, p. 5, 1889, 
