Mr. Herschel on various points of Analysis. 461 
even one or more arbitrary functions. A nearer attention to 
every step of the above reasoning will explain this paradox. 
But what has been said will serve to make us cautious in trust- 
ing implicitly to all its other applications. 
Problem I. To determine (p {x) from the equation (p" (x) 
= X. Assume z a function of x, and u a functional charac- 
teristic, which shall satisfy the following conditions 
j: = u , p (x) z= u , . 
From these, we obtain 
(p (x), that is, (p {uf) or {<pu)^ = ; {a) 
and (p' (x) or (p = x = u^; . . . • (6) 
and, subtracting, ~ ~ 
that is, A I {<P^)z + ^ 
and integrating, 
0 = + C!. 
Now by cross-multiplication of the equations (a) and (6) we 
find, 
u . . {(pu) , , — u . ((pn)^- 
Thus the function u . ((pu)^ does not vary when 2 ; changes 
to ^ 1 , and of course must be considered as constant in the 
integration of (r). C therefore may be any function of 
and thus our equation becomes 
0=“^+ 
oro = x + <p (x) +/ |x . (? (x)| 
an equation from which p {x) may be obtained for any as- 
signed form of the function /. Thus if/ [x) = a bx, 
mdcccxiv. 3 O 
