462 Mr, Herschel on vaiious points of Analysis, 
0 a "j“ X -j- bx'^ . (p (^x^ 
and <p(z) = -{^ 
which satisfies the condition proposed ; and by giving/ other 
forms, we should obtain other values of <p (x). 
The subsidiary function z, and the characteristic u are not 
then necessary to be known but as a matter of curiosity. They 
may however be found when (p is determined, by the resolu- 
tion of the equation of differences (p {uf) = gives 
the form of the function in z, and z is given by the equation 
X = or Z =: (x). 
Aliter. Assume as before, x = cp (x) = 
then we have 
(v.) = “*’ 
Now, X = ii^ therefore <p (x) = cp (w^), that is, = (p 
and for z writing s; — 1, u = <pu which being substituted 
in i^d) gives 
Now this is a perfect function <p on both sides, and of course, 
taking the inverse function <p—^ on both sides 
u = u 
z-i-i *— 1 
whence, 
= C I cos 27 Tzj + (— 1)* . C' I cos Qvrzj 
C and C' being two arbitrary functional characteristics. Now 
= X, and consequently 
a* = C I cos 27 TZ I ( — 1 )* . C' I cos 27 T 2 : 1 . 
From this conceive z found in functions of x, and call it Z (x) 
then, 
