the calculus of functions. 
24a 
putting <p for x we have 
— 1 
r 2 „ dp fx __ dp > I dp x ] 
? ^ rfp * ” * ( * I 
—x -X 
- T ro a; dp fx 
or ?/'*• — = Tr 
Some particular solution of the original equation must now be 
assumed as the value ofj r s and the resulting differential func- 
tional equation must be solved. The. only particular case of 
the equation = -™- with which! am at present acquainted, is 
iqpy 7 — 3 i±V— 3 
. I i±\f— 3 \ 2 2 
Other more complicated equations containing the various 
orders of functions, and their differentials may be reduced to 
those of the first order by the same means, but great dif- 
ficulties still remain ; it is by no means easy to discover par- 
ticular solutions of the original equations, and even when these 
are found, the functional equations of the first order which 
remain to be solved, are of considerable difficulty. I shall 
therefore refrain from giving any more examples, and pro- 
ceed to show how functional equations involving definite inte- 
grals may be reduced to those we have already treated. Such 
equations might occur in a variety of curious and interesting 
enquiries, few of which have yet been noticed. D’Alembert, 
in one of the volumes of his Opuscules, has examined a ques- 
tion which may be referred to this class ; it is the following. 
Suppose a sphere composed of particles of matter, what mustbe 
the la w of attraction amongst these particles, so that the force 
of the whole sphere acting on a particle at a distance, may fol- 
low the same law ? the question might be varied by supposing 
the law to be given, and the form of the solid to be required ; 
