WHICH SATISFY - GIVEN CONDITIONS. 
117 
where [_d\ m is a linear function of n, m, z, or, what is the same thing, of m, A, z. I 
assume for the moment that when the coefficients are determined [«] m would be found 
to have the value =[«]. 
Similarly, if ( a , b) m denote the number of the curves C r which have with the given 
curve U m contacts of the orders a and b respectively, and which besides pass through 
^r(r-\-3)—a—b points, then if the given curve break up into the curves m, m', then we 
have 
(a, b) m+ml -(a, b) m —(a, b) m/ ={(a) m (b) mf } + {(a) ml (b) m }, 
where { (a) m (b) m i\ is the number of the curves C r which have with m a contact of the 
order a and with m! a contact of the order b, and which pass through the \r[r- )-3) — a — b 
points; and the like for \{a) m \{b) 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 
{(«).(»)-}=(«).•(*)-= H-IAU 
{ }=(«}-•©»= 
and thence the functional equation 
{a, b) m+ml —(a, b) m —(a, b) ml =[a] m [b] ml +[a] ml [b] m . 
But \a] m &c. being linear functions of m. A, z, we have 
M m+ml M.+[«U p] m+mJ P1.+P] ml 5 
and thence a particular solution of the equation is at once seen to be [a] m p] OT ; the 
general solution is therefore 
(a, b) m =[a] m [b-] m +[a, b\, 
where [a, b~] m is an arbitrary linear function of m, A, z. Hence, assuming for the pre- 
sent that if determined its value would be found to be =[a, 5], we have the required 
formula ( a , 5)=[«][5]-|-[a, b~\. 
The investigation of the expression for (a, b, c) m depends in like manner on the 
assumption that we have 
{{a)Jb, c)*}={a) m .{b, P, 
and so in the succeeding cases ; and we thus, within the limits in which these assumptions 
are correct, obtain the series of formulae for (a, 5), (a, b, c ) 
85. It is to be observed in the investigation of (a, b) that if o—b , the two terms 
[a] m p] w and [_a] m i\J)] m become equal, and the equal value must be taken not twice but 
only once, that is, the functional equation is 
(a, a) m +w— (a, a) m ,=[a] m [a] m „ 
and the solution, writing \[_a, d] m for the arbitrary linear function, is 
(a, a) m =l[a\[a] m ^\[a, d] m , 
in which solution it would appear, by the determination of the arbitrary function, that 
[«, a\ has the value obtained from [a, 5] by writing therein b=ci. Writing the equa- 
tion in the form 
(a, «)=ip][a]+£|>, a], 
s 2 
