406] 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
235 
The investigation of the expression for (a, b, c) m depends in like manner on the 
assumption that we have 
{( a )m(b, C) m '} = (a)m • (b, c) m ' = [&] m {[6], n ' [c] m > + [&, 
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, b), (a, b, c).... 
85. It is to be observed in the investigation of (a, b) that if a = b, the two 
terms \a\ m and [a] m - [6] m become equal, and the equal value must be taken not 
twice but only once, that is, the functional equation is 
(it, Cl)m+m' (yj u)m (u> Ct)rn' — [«] m 
and the solution, writing \ [a, a] m for the arbitrary linear function, is 
(ii, Cl) m — \cC\m [®]m + o’ [®> &]?»> 
in which solution it would appear, by the determination of the arbitrary function, that 
[a, a\ has the value obtained from [a, by writing therein b — a. Writing the 
equation in the form 
(a, a) = \ [a] [a] + £ [a, a\ 
and comparing with the equation for {a, b), we see that [a, b] is not to be considered 
as acquiring any divisor when b is put = a, but that the divisor is introduced as a 
divisor of the whole right-hand side of the equation in virtue of the remark as to 
the divisor of the functions (a, b), (a, b, c)... in the case of any equalities between 
the numbers (a, b, c...). This is generally the case, and the foregoing expressions for 
[a, b], [a, b, c], &c. are thus to be regarded as true without modification even in the 
case of any equalities among the numbers a, b, c.... 
86. To complete according to the foregoing method the determination of the 
expressions for (a), (a, b),.., we have to determine the linear functions [a], [a, 6], &c., 
which are each of them of the form fm + gk + h/c, where (/, g, h) are functions of r 
and of a, b, &c.; and I observe that the determination can be effected if we know 
the values of (a), (a, b), &c. in the cases of a unicursal curve without cusps and 
with a single cusp respectively. Thus assume that in these two cases respectively 
we have 
(a) = (a -f 1) (rra — a), 
(a) = (a + 1) (rra — a)-a. 
Writing first A = — 1, k — 0, and secondly A = — 1, k — 1, we have 
(a + 1) (rm - a) =fni-g, 
(a+ 1) (rm -a)-a= fm —g + h, 
whence 
/=(a+l)h g = (a+l)a, h = -a, 
30—2