calculus of functions and other branches of analysis. 203 
in which the solutions vanish from some particular values of 
the coefficients, all equations which are homogeneous relative to 
the different forms of the unknown function are comprehended in 
it, as are also all equations which are symmetrical relative to the 
same quantities. 
There exists another class of equations nearly allied to those 
which are symmetrical relative to the different forms of the 
unknown functions, whose solution I shall now point out, 
chiefly with the view of giving another example of a method 
of reasoning which may frequently be employed with advan- 
tage in these inquiries, and also for the purpose of mention- 
ing a remark respecting the elimination of variables in a cer- 
tain class of algebraic equations which I do not recollect 
to have seen noticed. The class of functional equations to 
which I allude, are contained in the expression 
F | -tyx, ypax, . . 4 >a n ~~ l x, x, ax, .. a n ~ l x j = 0 
which for the sake of convenience may be written thus, 
F | -tyx, -fyux, . . ■tya n ~ 1 x,fx,fx,fx, Sec. j = 0 
where a n x = x and fx,fx,fx, &c. are any symmetrical func- 
1 2- 
tions of x, ax, . . a n ~ r x. 
In this equation none of the known functions fx, fx. Sec . 
are changed by the substitution of ax, u*x, . . . a l ~ l x for x ; 
and since the form of the unknown function depends on that 
of those which are known, it follows that the form of the 
unknown function will not be changed by the substitution of 
ax, a*x, . . a n —'x for x, or in other words we may suppose 
ipx = ipax = Sec. = ‘lia n — l x, and consequently >}/# will be de- 
termined by the equation 
F tyx, . . . 4 >x,fx,fx,fx, Sec. j = 0 
