422 Mr. Babbage's essay towards 
order may satisfy any number of conditions (which are not 
contradictory) simultaneously; and there appears no reason 
for denying this property to those of higher orders. 
If now we consider the symmetrical function 
% | fe 4 / * a?, . • \}/ a? j- 
and if x — x. It is evident this function will not change 
by the substitution of x, ^ x, Sec. or x 
and consequently that a different function similarly consti- 
tuted may be substituted for each of the arbitrary quantities 
a, b, c, Sec. in ( 1 ) which is the solution of the equation 
F | x, \J/ x, 4 /* x, . . \|/ n x J = o ( a ) 
The number denoted by u is arbitrary (it may, however, be- 
come determined from some particular circumstances of the 
equation (a). 
Thus we have introduced into an equation of the w th order, 
an unlimited number of arbitrary functions, each of which 
contains the function whose determination was sought with 
all its different orders to an undefined extent. 
If we take the particular case of 
4* x = a — x 
^ x = X 
u must be unity, and a general solution is 
taking another solution 
and 
If X= x v=n — l , and supposing -tyx =f (x,a,b, Sec . ) 
any particular solution we have for the general one 
