2 6 
Mr. Herschel on the 
would avoid their use, much circumlocution, as well as very 
unwieldy formulae must be introduced. Arbogast also, in 
his work on derivations, has given two most ingenious 
demonstrations of them, and added greatly to their gene- 
rality ; and lastly, Dr. Brinkley has made them the subject 
of a paper in the Transactions of this Society,* to which I 
shall have occasion again to refer. Considered as insulated 
truths, unconnected with any other considerable branch of 
analysis, the method employed by the latter author seems 
the most simple and elegant which could have been devised. 
It has however the great inconvenience of not making us 
acquainted with the bearings and dependencies of these 
important theorems, which, in this instance, as in many 
others, are far more valuable than the mere formulas. 
The theorems above referred to are comprehended in the 
equation. 
where the A applies to the variation of x, and the D to the 
functional characteristic u; and where n may have any value 
whatever. 
Taking these theorems for granted, I shall observe, that, 
in their present form, they are but abridged expressions of 
their meaning, and that to become practically useful, their 
second members must be developed in a series of the powers 
of Ax.D. This part of their theory has been most beautifully 
and satisfactorily treated by Laplace in the case of n = — - 1 
or, more generally 
* Phil- Trans. $807. 1 , 
