52 ON THE THEORY OF ALGEBRAIC CURVES. [10 
employed, therefore, even when several consecutive terms of the series t, t' are 
equal. It will be convenient also to assume that p — in is not negative, or at least 
for greater simplicity to examine this case in the first place. 
u, U, and v, V, contain terms of the form x a yPU, xvy s V, a + (3 :}> p — m, y + 8 p — n; 
jL)II contains terms of this form, and in addition terms for which a + /3=p + l — m, 
ry q- 8 = p +1 — n. It is useless to repeat the former terms, so that we may assume for p, 
a homogeneous function of the order p — m — n + t +1, or p — m+ 1 ; in which case jjII 
consists only of terms for which a + /3 = p + 1 — m, y + 8 = p + 1 — n. And the general 
expression of p contains p — m — n + t + 2, or p — m + 2, arbitrary constants. Similarly 
p II' contains terms of the form of those in uTJ, vV, pTl, and also terms for which 
a + /3 = p + 2 — m, y+8 = p+2—n) the latter terms only need be considered, or p may 
be assumed to be a homogeneous function of the order p —m — n + t' + 2, or p — m + 1, 
containing therefore p-m-n — t'+ 3, or p — m + 2 arbitrary constants. 
Similarly p lk ~ 1} contains p — m — n + t {k ~ 1] + k + 1 or p — m+2 arbitrary constants. 
Assume 
fp — m — n + t + 2\ (p —m — n + t' + 3\ (p — in — n + + Jc + 1\ n 
^ \p — m + 2 /~\p — m + 2 \p — m + 2 ) 
where, in forming the value of V the least of the two quantities in ( ) is to be 
taken ; this value also, if negative, being replaced by zero. The number of arbitrary 
constants in p, p' ... g) {k -v is consequently equal to V. 
The numbers of arbitrary constants in a, v, are respectively 
[1 +2 + (p — 771 + 1)} and [1 +2 + (p — n+ 1)} 
i.e. \ (p - in + 1) (p - m + 2), and |(p — n + 1) (p - ti + 2) ; 
thus the whole number of arbitrary constants in W, diminished by unity (since nothing 
is gained in generality, by leaving the coefficient (for instance of if ) indeterminate, 
instead of supposing it equal to unity) becomes 
(p — in+ 1) (p — m + 2) + i (p — ii + 1) (p - 11 + 2) + V — 1, 
reducible to 
|p (p + 3) + £ (p — 777- - n + 1) (p — 7?i — n + 2) — inn + V. 
By the reasonings contained in the paper already referred to, if p + k -in - n+\ 
be positive, to find the number of really disposable constants in W, we must subtract 
from this number a number | (p + k - in — 11 + 1) (p + k — in - 71 + 2). Hence, calling </> 
the number of disposable constants in W, we have 
0 = 2P (P + 3) + 2 (P ~ m ~ 11 + 1) (P ~ 171 — 71 + 2) inn + V - A (11), 
where A = 0, if p + k — in -77 + 1 be negative or zero (12), 
A — | (p + k — m — 71 + 1) (p + k — in — 71 + 2), 
if p + k — in — n+1 be positive; and y is given by the equation (9).