the calculus of functions. 
413 
If AB = x and the equation of the curve be y = ^x. 
PB =y = and AC = PB = v|/<r, 
and OC = T q, x = q/ x, and generally the n th ordinate TF is 
equal to T” x, hence 
x — x 
which is the equation whose solution has been just found. 
Problem XII. 
Given the equation 
\J ? x = 06 X 
required the form of T. 
Assume q ,x = <p t f <p x 
then q/ x= (p /* (p x 
and <p f 1 <p x = x x 
take the function <p on both sides, then this becomes 
f* <p x = <p a. X. 
This is a functional equation of the first order relative to <p, 
and may be solved either by the methods exhibited in the first 
part of this paper, or by the very elegant one of Laplace, 
f is a perfectly arbitrary function, except that neither fx nor 
f° x must be equal to x : from not attending to this circum- 
stance,! was at first led into several errors ; the reason of these 
two restrictions is, that in the first case we at once determine 
\(/cr to be equal to x, and in the second, we in fact make 
ocx = x, neither of which are necessarily true. 
Problem XIII. 
Given the equation. 
x = oc x 
This admits of a solution similar to the last, by assuming \J/a? 
