the calculus of functions. 41 1 
in reality, for it may easily be shown, that from the sum and 
product of two quantities any symmetrical function may be 
composed. 
As particular cases of the equation 
\{ r oc — x 
we may notice 
^x = a — x, and ‘LL±^±JjEz±fl 
2 
Problem X. 
Required the solution of 
•v{/ e x = oc 
another solution may be found from the following principle. 
Assume ^ oc = q> f <p x 
hence V oc = <p f q> <p* f <p oc = $ / 2 <p x 
or <p' f* (poc = oc. 
Now this equation can be satisfied if we are acquainted with a 
particular solution of / 2 x = oc, this may be found from the 
last problem, and since/ 2 x = x, the equation becomes 
— 1 
(p (p x = x 
which is identical, consequently 
\|/ x = (p f <p x 
some particular cases are 
(a -*«,). ?'(^)&C. 
from each of these by assigning particular values to <p, new 
values of f may be determined, and these in their turn will 
furnish new forms of the function -i/oc. 
Some time ago I received from the gentleman already al- 
luded to, the following solution of the equation 
OC=OL 7 
\|/ x = (p ( — q>oc) 
3G2 
