20 6 
Mr. Babbage on the analogy between the 
again u m a m . . . [n ) oc = a n x m = a n a n . . . ( w) < 3 ? =±= 
— a n a n ... ( m — -1 ) <37 = &C. = a. n u n X = a n X = X 
consequently whenever m is a whole number a m x is a solu- 
tion of \J/ W <2? = <37. 
If we take the particular case of n = 3 we have %|/ 3 <37 == <37 
and ux — — — is one of its solutions, therefore other solutions 
will be ax ■=—?—. olx = — — cc 1 x — x. 
These expressions when generalised by the introduction of 
an arbitrary function, do not give solutions which are irre- 
ducible to each other, nor do they even then in my opinion 
contain all possible solutions ; by introducing an arbitrary 
function they become 
the latter of which gives if we make <p<37 = 
If we consider the equation •i/ i x = x one of its values is 
ax = this gives for the others. 
All these forms will satisfy the equation ^*<37 = x and they 
may all be generalised by the introduction of an arbitrary 
function similar to that employed for the equation ^37 =,3?, 
but I do not suppose these solutions when thus generalised 
would contain all possible ones. 
Perhaps the following observations may throw some light 
on the generality of the solutions of such equations as those 
we are now considering. 
In the first place, it is obvious that every solution of $ 3 x—x 
