Mr, Herschel on various points of Analysis, 463 
I = (p (x) = C { cos 27tZ (x) } - (~ if . C' { cos 27 tZ (x) } . 
This method applies also to the more general equation (p^ [x) 
(-^)> by ^be substitutions f (jr) ^ u , p [x) = ,but, 
owing to the transcendental equations it introduces, must be 
regarded as totally ineffectual and useless. 
Prob. II. Given (p" (x) = f {x). Required at least one 
satisfactory value of p (x). 
Let the general expression ofy* (x) in functions of z and x 
found according to the method above explained be F x|: 
I 
we have then p^ =zj\ and p ==/", that is 
I 
<p {x) (x) = F {7.x}. 
Ex. 1. Let/(x) = ax'* — 1, or p^ [x) = 2X^ — 1, and we find 
f (x) = ^ + {x — lY } 
and of course 
(x) = ? (X) = i { (X + + (X - } . 
We may here observe that a?iy one of the n values of”V2 will 
equally afford a satisfactory value of p (x). 
Ex. 2. Let / (X) - =/(x) 
Assume tet = x : 
and we shall find 
/" (x) or <p (x) = 
+ (« — I • ' ” — p + X I 
I 
— (cT+/tA)| .1-” — I car 4- («r— / a) ^ 
where any of the n values of i/” may be taken, and thus as 
3 O 3 
