ON THE THEORY OF ALGEBRAIC CURVES. 
51 
10] 
o! — A\ ¡3' = B 6' = 11' are not all of them satisfied simultaneously). It will be 
seen presently that we shall gain in symmetry by wording the theorem thus: the 
degree of II is equal to the greatest of the two quantities m + n-t- 1, w-1. 
Suppose next, in addition to the equations (8), we have 
a! = A', ¡3' = B' f' = F, t’ equations, t! > t... (8'). 
Then, taking T', T', P', Q' the analogous quantities to T, % P, Q, and putting 
we have, as before, 
EV. U-ET. V = IT, 
IT = E [ET. PT'. (P' - Q') + PT'. P'AT' - ET. Q'AT'}. 
The degree of P'— Q' is t'- 2 (unless simultaneously a" = A", (3" = B" f" = F\ in 
which case the degree may be lower). The degrees, therefore, of the terms of IT are 
m + n — t' — 2, n — 1, m — 1. Or we may say that the degree of IT is equal to the 
greatest of the quantities m + n-t' - 2, m - 1 ; though to establish this proposition in 
the case where t' = n — 1 would require some additional considerations. 
Continuing in this manner until we come to the quantity II ( * -1 >, the degree of 
this quantity is the greatest of the two numbers m + n — t (k ~ 1} — k, m — 1. And we 
may suppose that none of the equations a (k) — A {k) are satisfied, so that the series 
n, IT IT*- 1 » is not to be continued beyond this point. 
Considering now the equation of the curve passing through the mn — t — t'... — t ik ~ 1} 
points of intersection of U = 0, V= 0, we may write 
W — uU + vV+pll +p'IT -fpifc-D n (t_1) = 0 (9), 
for the required equation; the dimensions of u, v, p, p'...being respectively 
p — m, p — n\ p — m — n + t+lovp — ra+1; 
p — m— n+ t’ + 2 or p —m + 1; p — m— n + ¿ (ifc_1) + k or p — m + 1, 
the lowest of the two numbers being taken for the dimensions of p, p' ...p (k ~ 1] . Also, 
if any of these numbers become negative, the corresponding term is to be rejected. 
In saying that the degrees of p, p' p( k ~v have these actual values, it is supposed 
that the degrees of II, 11' II* -1 actually ascend to the greatest of the values 
p — m — n + t + 1, or m — 1; m + n — t' — 2, or m — 1 ; — m + n — t (k ~ 1} + k, or m — 1. 
The cases of exception to this are when several of the consecutive numbers t, t' 
are equal. In this case the corresponding terms of the series II, IT IT* -1 ', are 
also equal. Suppose for instance t, t' were equal, II, IT would also be equal. A term 
of p of an order higher by unity than p — m — n + t + 1, or p — m+1, which is the 
highest term admissible, produces in pH a term identical with one of the terms of 
p'II; so that nothing is gained in generality by admitting such terms into p. The 
equation (9), with the preceding values for the dimensions of p, p' p (k ~ l) , may be 
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