41 6 Mr. Babbages ’s essay towards 
take successively on each side the functions (p,f and <p, the 
equation becomes 
a(y, c P'f ( Py) — &f- I( Py 
for y put (p 1 y, then 
* (<p'y> Q fy) = ( P t f-*y, 
this is a functional equation of the first order relative to <p , 
give/ any determinate value and solve the equation. 
From hence we may deduce the solution of the following 
question. 
Required the nature of a curve 
such that taking any abscissa AB and drawing the ordinate 
CB, if with centre A and radius AC, we describe a circle cut- 
ting the abscissa in D, the ordinate ED may be equal to the 
first abscissa AB. 
Let y = %{/ oo == CB 
AD = CA = \/ a? -j- (\}/ x)" 
and ED = -p (AD), hence 
*==^( v /F+l+I)' s ) 
which is a particular case of the preceding problem. 
Problem XVIII. 
To reduce the equation 
F {x, -tyx, \}/ 2 ux , . . -p n v a?) = o 
