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the calculus of Junctions. 
inclined to think that the solution of functional equations must 
be sought by methods peculiarly their own. There are some 
other researches on this difficult subject of which I am un- 
able to give any account, from the impossibility of procuring 
the works in which they are contained ; among these is the 
paper of Arbogast, which gained the prize of the Academy of 
Petersburg in the year 1790. 
For the sake of convenience, I shall call any solution of a 
functional equation which contains one or more arbitrary 
functions, a general solution ; but if the solution of such an 
equation only contains arbitrary constants, I shall call it a par- 
ticular solution. With respect to the number of arbitrary 
functions that may enter into any solution, I shall make some 
observations at the conclusion of this paper. 
Problem I. 
Required the general solution of the functional equation, 
4* X = a X 
supposing we are acquainted with one particular solution. 
Let the particular solution be fx =^=/ u x ; then take ip = <pf, 
<p being an arbitrary function. It is evident that this value 
of \p will satisfy the original equation, and that 
<pfl V — pj a X 
is identical, because/x =/ax* 
Example, let the equation be 
M = '!' (— x) 
and the particular solution be 
fx =af 
the general solution is 
4^ x == <p (x 2 ) 
which evidently answers the conditions. 
3 E 2 
