1206 
MR. R. F. GWYTHER OR THE DIFFERENTIAL 
the roots in the latter case being all real, provided 
256U73 - 243 %® > 0 . 
28. Case of cuspidal cubic. 
The ratio (p/f is now to be determined from the equations of condition, as well 
as xjj/f 
Writing u^^(f)lf = q, we may write down the conditions from that obtained for the 
nodal cubic, viz., 
(P + qf — Su^Jc = 0. 
(F- + ^)^ — I {F + g) + ^ 8 - ^^5® = 0 . 
Whence, 
256q^= 
IGF = 27 ig^ I 
^ j 
and the differential equation becomes 
256 U 7 ® -27%® = 0.. (54). 
29. To exemplify the use of the general forms of the equation to the cubic and of 
the matrix. 
The coordinates of the tangential of the origin are 
TT 
= 0, ^=- 
a 
and the matrix of the polar conic of the tangential becomes 
{ 4 >i + 2^2Lq) [/?q +{(() + + (/)3Lg2)] —fxfj^. 
Taking the general cubic this becomes 
(W+ 2Vf,) 
^7% + 
- ¥ 3 % - jjf/y 
+ u 
65 
and the matrix(and conic)breaks ujd into factorSjj^rovided %= 0,sinceU7^ —VgLg —U 7 Lg" 
contains the factor 
That is, as is known, the tangent at a sextactic point passes through a point of 
inflexion. 
The matrix of the nodal cubic is given by 
+ L„ + H - ^ 
% 
