4*2 
Mr. Babbage’s essay towards 
and likewise the solution <p' (( — 1 )"cpx) of the equation 
■J/ n x = x; this first led me to the substitution of <p '/ <px, which 
is of such essential use in these enquiries. 
Problem XI. 
Given the equation. 
x —x. 
Assume as before -\ix = <p ' ftp x, 
then 4* x = <p fp p 'fpx = p */ e p x 
^ 3 x ==<£'/* <p p 1 fpx = p'f 3 px, 
and generally -fy n x = <p * f n <p x, 
hence our equation becomes 
<p f n <p x = x. (tf) 
Suppose we have one particular solution cf the equation, 
substitute this instead of /, and the equation [a) becomes iden- 
tical, whence 
■fyx — p 1 fpx 
and from this other values of/ may be determined, and so on 
ad infinitum. 
The equation we have just considered, affords a ready solu- 
tion of the following Problem. 
Required the nature of a curve, such that taking any point 
B in the abscissa, and drawing the ordinate BP 
if we make AC another abscissa equal to BP the preceding 
ordinate, and if we continue this n times, then the ?V- h ordinate 
may be equal to the first abscissa. 
