248] 
ON THE THEORY OF INVOLUTION. 
311 
which last equation, to denote that it is of the second order in regard to the differential 
coefficients a, b, &c., a', &c. I represent by 
((a, b, ...) 2 $X, fi, v) 2 = 0. 
But this system of four equations contains not only the cusp system, but the 
system made of the three linear equations and the equation x 2 = 0. Klim mating 
X, n, v, the last-mentioned system is 
= 0, x 2 = 0, 
L, 
L', 
L" 
M, 
AI', 
AI" 
N, 
N', 
N" 
Avhere the first equation is that of a curve of the order 3 (n — 1). And the two 
equations give together 6(n — 1) points, viz. the points of intersection of the curve by 
the line x = 0, each reckoned as a twofold point. 
Returning to the first-mentioned system, this may be replaced by 
L" = o, ({ a , b,. - L"M', L"AI— LAI", LAI' - L'Al) 2 = 0, 
M" 
N" 
which are curves of the orders 3 (n — 1) and Qn — 8 respectively. But each of these 
curves passes through the 3(?i — l) 2 points given by the equations 
I, L, L', L" 
\ AI, M, AI" 
L, 
L\ 
AI 
M 
N 
N' 
= 0, 
and these points are moreover nodes on the curve of the order Qn — 8; hence the 
points in question reckon as 6 (n — l) 2 intersections of the two curves. The number 
of the remaining intersections is 
3 (n -1) (Qn - 8) - 6 (n - l) 2 = 6 (n - 1) (3n - 4 - (n - 1)) = 6 (n - 1) (2n - 3), 
but among these are included the 6(n — 1) intersections of the curve of the order 
3 (n — 1) by the twofold line x 2 = 0; or, subtracting these, the number of the remaining 
points is 
6 (n — 1) (2n — 3 — 1) = 12 (w — 1) (A — 2) ; 
which number is consequently the order of the cusp system. 
It may be remarked that considering the entire series of equations at first denoted 
by (L = 0, AI = 0, JSf = 0), (A = 0, B = 0, (7=0, F — 0, (7 = 0, II — 0), the elimination of 
X, v from the three linear equations gives as before 
= 0, 
L, 
L', 
L" 
AI, 
AI' 
AI" 
N 
N' 
N"