310 
ON THE THEORY OF INVOLUTION. 
[348 
a system of equations which is contained in the system, L = 0, M = 0, N = 0, A = 0. 
But this system contains besides the cusp system the irrelevant system L = 0, M — 0, 
A =0, x 2 = 0. In fact the equations L = 0, M=0, N = 0 give 
ax + liy + gz = 0, 
lix + by +fz = 0, 
gx +fy +cz = 0, 
and thence 
x- : y- : z 2 : yz : zx : xy = A : B : O : F : G : H. 
Hence the equation A = 0, if x 2 = 0, implies the entire system 
A = 0, B = 0, 0 = 0, F = 0, G = 0, J/ = 0. 
But if L = 0, M = 0, N = 0, ¿r 2 = 0, then these equations give A = 0 (and also H = 0, 
G = 0), but they do not give the remaining equations B = 0, G=0, F = 0. Or the 
same thing may be shown in a less symmetrical form, but more clearly thus; we 
have identically 
— cx (ax + hy + gz) + (hx — fz) (lix + by+ fz) + bz (gx + fy + cz) — (be — / 2 ) -s 2 + (ah — h 2 ) xr = 0, 
whence the equations L = 0, M= 0, N = 0, A = 0 give Gx 2 = 0, that is, 0=0, or x 2 = 0. 
But the equations L = 0, M=0, JSi = 0, A =0, G = 0 give (as it is easy to show) the 
entire system A = 0, B = 0, C=0, F = 0, G = 0, H=0. That is, the system 
¿ = 0, M= 0, N= 0, A = 0 
is made up of the cusp system, and of the system (L = 0, M = 0, N = 0, A = 0, oc 2 = 0); 
or since A = 0 is a consequence of the other equations, the second system is 
(L = 0, M= 0, N = 0, x 2 = 0). 
Consider now the curve \ TJ + g JJ' + v TJ" = 0, which will have a cusp if the ratios 
X : ¡i : v are properly determined. And to each set of values of \ : g : v there corre 
sponds a set of values (x, y, z), the coordinates of a cusp of the curve; so that the 
number of such sets, that is, the number of points each whereof is the cusp of a 
corresponding curve \U + gV + vW = 0 is precisely equal to the number of sets of 
values of X : g : v : or it is equal to the order of the system of conditions for the 
existence of a cusp. 
Denoting as before the first and second differential coefficients of U by 
L, M, N, a, b, c, f g, h, 
and those of U', TJ” in a corresponding manner, and taking for the cusp the system 
before represented by L = 0, M =0, N = 0, A = 0, we have 
XL + gL -f- vL = 0, 
XM + gM' + vM” = 0, 
XN + gN' + vN” = 0, 
(Xb + gb' + vb”) (Xc + gc' + vc”) — (Xf + gf' + vf // ) 2 = 0,