260 
ON THE THEORY OF INVOLUTION IN GEOMETRY. 
[40 
the theory, of considerable interest, to the problem of elimination between any number 
of equations containing the same number of variables. Suppose, for instance, two equa 
tions, TJ = 0, V = 0, when U, V are homogeneous functions of x, y of the degrees m, n 
respectively. To eliminate the variables it is sufficient to multiply the first equation by 
x n ~ 2 y..., y n ~ x , and the second by y m_1 , and from the equations so obtained 
to eliminate linearly the (m + n) quantities x m+n ~ 1 , x m+n ~ 2 y..., y m + n ~\ But in the case 
of a greater number of equations it is not at first obvious how many new equations 
should be obtained; and when a number apparently sufficiently great have been found, 
it may happen that the equations so obtained are not independent, and that the elimi 
nation cannot be performed. But in showing the connexion that exists between these 
different equations, the theory of involution explains in what manner a system is to be 
formed, which includes all the really independent equations, and gives the means of 
detecting the extraneous factors which appear in the result of the linear elimination of 
the different terms; but I do not see at present any mode of obtaining the final result 
at once in its reduced form free from any extraneous factors. 
Let X, Y, ... be given homogeneous functions of the same degree of any number of 
variables, and suppose 
® = aX + /3Y+..., 
a, /3... being constants, and the number of terms in the series being g; © contains 
therefore g arbitrary constants. If however, by giving to a, /3 ... particular values 
a lt ¡3 l ..., or a 2 , (3 2 ..., and representing by © x , © 2 ... the corresponding values of ©, we 
have identically 
©! = 0, © 2 = 0,... (h equations); 
then the constants in © group themselves together into a smaller number g — h of 
arbitrary constants. This supposes, however, that the last mentioned equations are linearly 
independent; if there are a certain number Jc of equations 
dfi = 0, d> 2 = 0 ..., 
(where d> l5 d> 2 , ... are linear functions of © 1} © 2 ...) which are identically satisfied, inde 
pendently of the h equations, then the equations in question are equivalent to h — k 
equations, and the function © contains g — (h—k) or g — h + k, arbitrary constants. 
Similarly if the functions <I> are not independent; so that the number of arbitrary 
constants really contained in © is always 
X — g — h + k — &c. ... 
Consider now the case of a function ©, homogeneous of the r th degree in the variables 
x, y...{(6 +1) in number}. Let U, V... be functions of the degrees m, n..., and suppose 
® = uU +vV + ... 
where u, v ... are arbitrary functions of the degrees r — m, i— n, ... [r is supposed 
throughout greater than m, n ...}. Suppose for shortness that the number of terms in 
the complete function of 9 variables, and of the order p, i.e. the 
quotient 
[P + QY 
is