in Mathematical Reasoning. 345 
The principle of representing any one quantity indifferently, 
out of a given number, has been employed on several occasions, 
and occurs in some mechanical questions. Lagrange has made 
use of it in a memoir relative to the theory of Sound*, and 
M. Poisson has employed it on a similar occasion +. The cause, 
on which its successful application depends, seems to be the 
power which it gives of uniting together a number of cases 
totally distinct, and of expressing them all by the same formula, 
the consequence of which is, that one single investigation fre- 
quently supersedes the necessity of a multitude. 
Many examples of the successful application of this princi- 
ple, are to be found in a paper on Circulating Functionst{, in 
which it is applied to the solution of a peculiar class of equations 
of finite differences: other mstances will come under our notice, 
in a future communication. 
An examination of the various stages, by which, from certain 
data, we arrive at the solution of the questions to which they 
belong, ought to constitute a prominent feature, in any work 
devoted to the philosophical explanation of analytical language. 
It is a subject, the consideration of which is too frequently 
omitted altogether, and as a correct view of it tends materially to 
advance the science, and also by pointing out more clearly the 
nature of the difficulties it has to contend with, and also by 
guarding us against those openings by which errors are most fre- 
quently introduced, I shall enter into the question at some length. 
In the application of analysis to the various questions which 
aye submitted to it{, there are three distinct stages, each subject 
* Memoirs de Turin, vol. I. + Journal de Ecole Polytecnique Cah. 18. 
{ Phil. Trans. p. 144, 1818. J.F. W. Herschel, Esq. 
§ Of course questions already in an algebraic form are not here alluded to, such as 
the integration of equations, &c. 
