calculus of functions and other branches of analysis. 207 
will also be a solution of tfx~x, such likewise will be all 
the solutions of ^ a x = x. The complete solution of -ifx = x 
should therefore contain all forms of x which satisfy the 
equations li/ 3 x = x and ^*x — x. Again, if a function as ax 
satisfies the equation ^x— >—x, it will also satisfy -fy 6 x—x, for 
since a 3 x = — x by putting a? x for x it becomes a 3 a 3 x = — 
a 3 x = — ( — x)—x, so also any function which satisfies the 
equation will fulfil the equation -ty*x = x which may 
be proved in the same manner. I shall however take the 
more general case, and show that if any function satisfies the 
equation ty 3 x=(3x where /3 is any function satisfying the equa- 
tion \J/ 2 x=x , it will also satisfy the equation •ty 6 x=x, for since 
a 3 x—/3x, putting a. 3 x for x, we have a 3 a 3 x~ax^= (3a 3 x~f2!3x= 
fi*x=x. In the same way it might be demonstrated, that the 
complete solution of -^ abc > (x)=x must contain all func- 
tions which can satisfy any of the following conditions 
yp a X = UX 
Ip a6 x = ax 
s 
&c. 
<- 
sf 
II 
II 
&c. 
&c. See. 
where a is any function satisfying the equation -ty bc ® c -(x)=x 
(2 ditto ditto ^ c -(a?)=a 7 
&c. See. 
ex, ditto 
B 
ditto $ cd &c -(x)=x 
/3 ditto 
B 
ditto $a d ®c.( X )=X 
&c. &c. 
Sec. 
The comparison of the integral calculus with that of func- 
tions will supply us with several very marked analogies, 
some of which promise when farther pursued to be of essen- 
mdcccxvh. E e 
