/ 
the calculus of functions. 
2 33 
and generally when a n x=zx the form of is determined 
by the equation. 
fa 
$ X =5 
fax 
fa + 
fa % X 
fax + 
fa z x + &c. 
+fa x 
fa X + fa 
It may be observed, that this method of discovering particular 
solutions by elimination, will not apply when the given equa- 
tion contains only the different forms of the function without 
the variable quantity itself : thus it is not applicable to the 
equation 
F | -Fr, -fycix, . . xpa n xj=o 
the reason of this is obvious ; for if we eliminate from this 
equation (by means of the n equations which arise by chang- 
ing the order in which the functions are placed), all the func- 
tions but 4/a?, we shall have a result containing nothing but %f/x 
and constant quantities, and therefore, ^x is equal to a con- 
stant quantity : it is true such a value of -tyx will satisfy the 
equation, but it scarcely deserves the name of a solution. 
Another exception is, when the equation 
F •[ X, ^X, ypaX, See. 4/ a n X j = 0 
is homogeneous relative to the different forms of the unknown 
function ; for in this case when we attempt to eliminate them, 
they all disappear together, leaving an equation of condition ; 
thus given 
= (a •— a?) a x and a 2 x = x 
we have \|/oa? == (a •— ax) ^ a 2 x = (a — ax) $x 
and -tyx = (a-—x) (a— ax) ^x or l = (a— ax) (a—a*x) 
which equation is not necessarily true. 
