261 
40] ON THE THEORY OF INVOLUTION IN GEOMETRY. 
represented by [p, &] ; then the function ® contains apparently a number 
([r — m, 0] + [r — n, 6~\ + ...) 
of arbitrary constants. 
But since we should have identically © = 0 by assuming u = LV, v = - LU, w = 0, &c.... 
(L the general function of the order r — m — n), or u = MW, v = 0, w = — MU (M the 
general function of the order r — m—p) &c., the number N must be diminished by 
[r — m — n, 6] + [r — m — p, 0] + [r — n — p, 6] + ... ; 
but the equations just obtained are themselves not linearly independent, and in conse 
quence of this the number of arbitrary constants has to be increased by 
[?— m — n — p, $]+...; 
and so on. Hence finally the whole number of arbitrary constants in the function 6 is 
N — [r — m, 6] + [?— n, 6] + [r —p, 0] + ... 
— [?— m — n, &\ — [r — m — p, 6] — [r — n — p, 6~\ — ... 
4[r — m — n—p, #]+...+ &c. &c (A). 
This however supposes that all the numbers i—m, r — n..., i—m — n..., are positive: 
whenever this is not the case for any one of them, the corresponding term is obviously 
to be omitted. With this convention the equation (A) gives always the correct number 
of arbitrary constants in ©: it will be convenient to represent it in the abbreviated 
form 
N = {r : m, n, p, ... : 6). 
An expression analogous to this, for the particular case of r = m, but incorrect on 
account of the omission of all the terms after the second line, has been given by 
M. Plucker (Crelle, tom. xvi. p. 55), and even some of his particular formulae are incorrect. 
But proceeding to examine some particular cases: if r>m + n+p+... — 0 — 1, then in 
the expression (A) either no terms are to be omitted, or else the terms to be omitted 
reduce themselves to zero, so that A r is given by this formula continued to its last 
term. It will be subsequently shown that in this case 
[r : m, n, p ... : 6) — [r, d] — mnp ... ; 
or in the case of two or three variables, we have the theorem, “If a curve or surface 
of the order r be determined to pass through the mn points of intersection of two 
curves of the orders m and n, or the mnp points of intersection of three surfaces of the 
orders m, n, p; then if r>m + n — 3, or r>m + n+p — 4, the curve or surface contains 
precisely the same number of arbitrary constants as if the mn or mnp points were 
perfectly arbitrary.” 
This is natural enough; the peculiarity is in the case where r ;j> m + n — 3, or 
r m + n +p — 4. For instance, for two curves, r m + n — 3, we have 
[r : m, n : 2} = [?— m, 2] 4 [r — n, 2] = [r, 2] — mn + [r — m — n, 2],