2i2 Mr. Babbage on the analogy between the 
impossible case : if we treat the equation -tyx = c\px n by his 
method, we shall find for its solution 
— log log a? 
^x — c Io s M 
and this solution satisfies the equation ■tyx = c-J t /x n indepen- 
dently of any particular value of n, and if we suppose n = — i 
— log 4 X 
we have -tyx = c lo S” 
for the solution of the equation %{/ x = c\ \ -J- whatever may be 
the value of c, and we have before shown that it cannot have a 
solution unless c = + 1. The only explanation I am at pre- 
sent able to offer concerning this contradiction, is one which 
I hinted at on a former occasion, viz. that if we suppose to 
represent any inverse operation which admits of several 
values, then if throughout the whole equation we always 
take the same root or the same individual value of it is im- 
possible to satisfy the equation, but if we take one value of J/ 
in one part, and another of the values of \J/ in other parts of 
the equation, it is possible to fulfil it by such means. This 
solution may perhaps appear unsatisfactory, it is however 
only proposed as one which deserves examination, and I shall 
be happy if its insufficiency shall induce any other person to 
explain more clearly a very difficult subject. 
One of the most extensive methods of integrating differen- 
tial equations, consists in multiplying by such a factor as will 
render the whole equation a complete differential, the deter- 
mination of this factor is, however, generally a matter of at 
least equal difficulty with that of solving the original equa- 
tion : analogy would lead us to suspect that some similar 
mode might be adopted for the solution of functional equa- 
