ON THE THEORY OF ALGEBRAIC CURVES. 
47 
10] 
the negative powers of x on the second side, in point of fact, destroying each other. 
Supposing in general that fx containing positive and negative powers of x, Efx denotes 
the function which is obtained by the rejection of the negative powers, we may write 
TJ = E (y — ax ... 
aim) ■ 
^¡=l) [y-fa.. 
n.m—1 ) 
y — \x... 
A< m >\ 
•(3), 
the symbol E being necessary in the present case, because, when the series are 
continued only to the power x w+1 , the negative powers no longer destroy each other. 
We may henceforward consider U as originally given by the equation (3), the 
m(m+l) quantities a, a'... a™, /3, ¡3'... /3™,... X, A'...A<™> satisfying the equations 
obtained from the supposition that it is possible to determine the following terms 
a m+l) , ... A (m+1) ,... so that the terms containing negative powers of x, on the 
second side of equation (2), vanish. It is easily seen that a, /3...A, a', ¡3' ...A' are 
entirely arbitrary, a , /3 ...A satisfy a single equation involving only the preceding 
quantities, a , /3 ...A two equations involving the quantities which precede them, 
and so on, until a' m) , /3 1 "' 1 ... A <m) , which satisfy (m—1) relations involving the preceding 
quantities. Thus the m [m + 1) quantities in question satisfy equations, or 
they may be considered as functions of m {m + 1) - \m (m - 1) = \m (m + 3) arbitrary 
constants. Hence the value of U, given by the equation (3), is the most general 
expression for a function of the ra th order. It is to be remarked also that the 
quantities a (m+1) , /3 ( " i+1, ... A <m+1) , are all of them completely determinable as 
functions of a, /3 ... A,... a (m) , /3 (m) ... A (w) . 
The advantage of the above mode of expressing the function U, is the facility 
obtained by means of it for the elimination of the variable y from the equation 
U = 0, and any other analogous one V = 0. In fact, suppose V expressed in the same 
manner as U, or by the equation 
(y ~ Kx • • • 
Kw\ 
a^'J 
(4), 
n being the degree of the function V. It is almost unnecessary to remark, that 
A, B... K, ... A (n) , B {n) ... K (n) are to be considered as functions of |n(n, + 3) arbitrary 
constants, and that the subsequent A {n+1) , B in+ v ...K {n+l) ... can be completely determined 
as functions of these. Determining the values of y from the equation (3), viz. the 
values given by the equations (2); substituting these successively in the equation 
V=(y-Ax-...)(y-Bx-...) ... (y-Kx-...) = 0 (5), 
analogous to (2), and taking the product of the quantities so obtained, also observing 
that this product must be independent of negative powers of x, the result of the 
elimination may be written down under the form 
E 
( ( . x a <mn) _jqm»n ( a mm) _ ]{(mn)\ 
|(a-4)a?...+ 1 ... j(a — Kx) ... + j 
x |(A — A') x 
\(mn) _ (mw > 
(X-Ka0...+ \ 
... (61