1204 
MR. R. F. GWYTHER ON THE DIFFERENTIAL 
These are sufficient to determine uniquely the coefficients in the matrix, and from 
the coefficient of we derive the differential equation to the non-singular cubic 
-/ + 
u- 
4>i — — ^ + = 0, 
u- 
which becomes 
«/U,U 5 - V,= - 4U/ + u*V, = 0, 
and is identical with that previously found in (36). 
And the matrix of the non-singular cubic is 
TT 2 _ V T, _ FT T 2 
- -A - + Vs -f U,L,. 
26. The matrix of the equation to the tangents to the cubic from the temporary 
origin. 
To find this equation, put tt = and form the discriminant of the resulting 
expression in ^ as a binary quantic. Equating this to zero, replace m by tt/^. 
Since we need only the coefficient of the highest power of tt, the simplest method of 
finding it is to put ^ = 0 in the matrix of the cubic, and form the discriminant of the 
resulting expression in tt. 
The matrix required is 
963^ — I26C3, 
or 
~ + V'iW) • • 
and the degree of this in Lg denotes the number of tangents which can be drawn. 
The conditions that the cubic may be nodal or cuspidal are that this matrix, as a 
function of Lg, may have a linear factor twice or three times repeated. 
27 . Case of nodal cubic. 
In this case we still determine uniquely all the ratios of the coefficients, except 
xp/f by the condition that the coefficients of h up to the seventh vanish identically, 
while the differential equation is found by equating the coefficient of to zero. 
The further condition, to determine xp/f is (47) that the discriminant of 
ifW + V'Lo - Aff - 4%Y (/Lo + rp) 
may vanish. 
Writing k or xpj2f and x for Lg + h, this equation becomes 
_ 2 (F -f U 7 ) .^2 - 4ifx -f (F U^)^ - Aufk = 0 . . . (48). 
