234 M r . Babbage’s essay towards 
Another exception is, when the given equation can be made 
to assume the form 
F ] -tyx, -fya x . . ’tycc n ac \ = fa- 
in this case the equation cannot be fulfilled unless fa is a 
symmetrical function of x, ax, &c. a n x, because the first side 
is such a symmetrical function : this reason, however, should 
be received with caution, for if the operation denoted by 4 be 
an inverse one, it may admit of several values, and it seems 
possible , that in such a case the condition relative to the form 
of f need not be fulfilled. In my former paper I explained 
the means of finding solutions of the equation fax~x. I then 
contented myself with explaining the theory without men- 
tioning particular cases; as these latter may be required in 
our present enquiry, I shall subjoin the following particular 
solutions of x=x 
fa=a — x fax=lo g (a — e*) fa =(a w — x n )~ n fa — ^i — ■ 
4^==^— fa=x — log(s*— 1) fa=z — - — 1 fa=^ 
X ~ 1 [ax n —\)n 
^=.-+3 
a — bx 
Particular cases of 4 3 a? = x 
4 *r=tan'(a— tanx) 
. . —I / sin (a — x) 
yx = tan — 
T \ cos a cos x 
fa = 
— fa = 
3 — X T 
V ax 2 - — a 1 
x 
fa = 
ac — c x 
fa: 
(ax n — a 2 ) w 
x 
^—1 
fa 
V x ’ 1 
fa 
a + bx 
b x -{-bc-\-c 2 
x 
fa = log (ae x — a 2 )- 
■x 
4® - ^=(^rp)” 4lX = log(?—l , )—X+C 
fa =.■£§; 4 , *=nb 4-x = — log ( 1 — f ) 
