COVARTAN'TS OF PLANE CURVES. 
1205 
With the usual notation (I and J) of theory of equations 
{¥ + 
(F + U ,)3 - f (F + U,) + ^3^ 
and the equation giving h is 
((P + u ,)2 - ^u.^hY - {(F + U,)3 - I {Jc^ + U 7 ) + ¥ = 0, 
a biquadratic for k. 
To find the differential equation we have 
U7^ = Vs/ 
and therefore 
31 
4 
8 
{(Vg^ + - 24?^,W3U,3]3 
-■[W+4Wf-3DU,WsU/(Y82+4U73) + 216w,sU/}2 = 0 . (49), 
which contains the factor 
The biquadratic from which k is to be found is 
+ + . . . (5 0). 
The discriminant of this is found to be 
161/58 {256U/ — 27%8]8. 
And if 256Uy8 — 27//58 > 0 , all the roots are real. 
In this case, four real nodal cubics can be drawn to have contact of the seventh order 
with the standard curve. If 256 U 78 — 27^8 < 0 only two. If 256 U 78 — 27u^ = 0 , 
there will be two nodal cubics, the third being cuspidal, as will appear in the next 
section. 
Equiharmonic and harmonic cubics can also be drawn to have contact of the 
seventh order, aad it is readily seen from the Theory of Forms that k will in these 
cases be determined by 
(P + - I u,^k (P + U7) + = 0 
and 
{k^ + = 0 . . . 
respectively. 
(51) 
(52) 
