calculus of frictions and other branches of analysis. 205 
If we apply this reasoning to algebraic equations containing 
several variables, as for instance to the two equations 
F (a:, y, a, b) = o, F (y, x, a, b) = 0 
we shall find that one set of the value of x is always con- 
tained in the equation F (x, x, a , 6 ) = 0. 
As an example, let us take the two equations 
y 2 -j- x — a and .z 2 -f-y — a 
one set of the values of x are contained in the equation 
x 2 -j- x—a = 0 
and this enters as a factor in the result of elimination, which 
gives 
a? 4 — 2 ax* -j- x ~\~a 2 — a— x — a) (a?- — x — a — > 1) = 0 
Another curious analogy exists between the calculus of 
functions and common algebra in the similarity of the rela- 
tions of the roots of unity to the solutions of the functional 
equation ^x — x. 
In the equation r n = 1 it is known that if r be one of the 
1 
roots then any power of r will also be a root, and if n is a 
prime number and r any root except unity, then r,r 2 , r 3 , . .,r w — 1 
x x 1 a 1 
will be all the different roots. Similarly in the functional 
equation ^x = a? if ux be one form which satisfies it, u m x will 
also fulfill it, and if n is a prime number then ux, a?x , . . 
will all be different forms which satisfy the equation. This 
may be readily shown as follows, if ux is a solution, then 
au .. ( n) X z=u n X=r-X 
Suppose m = 2 then 
u u . . (n) x = o? n x — u n ( u n x ) = u n x = x 
consequently ux is also a solution of ty n x = x 
