4^4 Herschel on various points of Analysis, 
many different values of (x) be obtained. This example 
depends on the integration of an equation of differences of the 
form 
0 z=u , .w4-A.w +B.I/4-C 
Z-{-\ Z ' JS-f-I ' z ‘ 
a particular case of which had been previously integrated by 
Laplace in the Journal de fEcole Polytechnique. 
Ex. 3. To take an instance of the application of these 
equations to geometrical pro- 
blems, let AM be an hyper- 
bola whose axis is CP and 
centre C, and let it be re- 
quired to find a curve am 
such that drawing the ordi- 
nate PMm, making Cd =z ^ 
Pm and again erecting dl 
parallel to PM, if this be repeated n times the last ordinate//^ 
shall be equafto PM. Lety = <p (x) be the equation of am 
andy = (1 — e~) [o' — x^) that of AM, then dl = (p {Cd) 
= (p {Pm) = y (a:) and in like manner, (p” (x) = PM, that 
is, (p” {x) =/ {x) = 1 — e'^) {al — P) ; consequently, 
1 i_ I 
f” ( j:) = (fi (x) = { (/ - 1 ) " .X” — 
Thus we see, that am is also an hyperbola, whose centre is C, 
and calling a' and e' its semiaxis and excentricity, we have 
e' = vX i)t + 1, and a' = X . { | *. 
If .^M be a right angled hyperbola, or ^ 2 , we shall 
have 2 and a* = ; that is, am is also a right an- 
gled hyperbola, having its axis part of that of AM. If e 
