234 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. [406 
and so on :, this is easily verified for (a, b), and without much difficulty for (a, b, c), 
but in the succeeding cases the actual verification would be very laborious. 
84. The theoretical foundation is as follows. Writing for greater distinctness (a), n 
in place of (a), we have (a) m to denote the number of the curves G r which have 
with a given curve U m a contact of the order a, and which besides pass through 
\r (r + 3) — a points. Let the curve U m be the aggregate of two curves of the orders 
to, to' respectively, or say let the curve U m be the two curves to, to', then we have 
(fG)m+m' ~ (®)m 4“ 
a functional equation, the solution of which is 
(u) m = [a] m > 
where is a linear function of n, to, k, or, what is the same thing, of to, A, k. 
I assume for the moment that when the coefficients are determined [«],„, would be 
found to have the value = [«]. 
Similarly, if (a, b) m denote the number of the curves G r which have with the 
given curve U m contacts of the orders a and b respectively, and which besides pass 
through \r (r + S) — a—b points, then if the given curve break up into the curves 
to, m, then we have 
(tt, &) w i+m' (®> b\ n ((I, b) m ' = !(u)m (b\n' \ 4" (b)ra\, 
where {(a) m (b) m is the number of the curves G r which have with to a contact of the 
order a and with to' a contact of the order b, and which pass through the \r (r + 3) — a — b 
points; and the like for {(a) m ' (6) m }. Then, not universally, but for values of a and b 
which are not too great, the order of the aggregate condition is equal to the product 
of the orders of the component conditions (ante, No. 12), that is, we have 
[W)>n = (®)m • (6)m' = [®]m [^]m' > 
X a )m j (®)m' • (by, n — [n] m / [&])«, 
and thence the functional equation 
(a, b^) m (a, b) m ' = [®J} № [6]»i' 4“ [u] m ' 
But [a\ m , &c. being linear functions of to, A, k, we have 
[a]m+m' = [®]m 4" [a]m', \b~\ni+m' = [^]j« 4“ [¿]m', 
and thence a particular solution of the equation is at once seen to be [a] wl [6] m ; the 
general solution is therefore 
(a, b) m — [<x]jn [^]m 4“ b~\ m , 
where [a, b] m is an arbitrary linear function of to, A, *. Hence, assuming for the 
present that if determined its value would be found to be = [a, b], we have the 
required formula (a, b) = [a] [6] + [a, 6].