G6 mk murphy, on elimination between an indefinite 



In the physical investigations, which conduct to an indefinite num- 

 ber of equations, it is of great importance to discover the law of those 

 quantities, corresponding to the law by which the given equations are 

 connected. The method which I here propose for this object is founded 

 on the two following principles. 



First, if we make the right-hand member of the a;*'' equation dis- 

 appear by transposition, the left-hand member is then a function of x, 



which vanishes when x is any number of the series 1, 2, 3, w; and 



therefore it must be of the form 



P.(.r- 1) {x-2) (x-S) (x-n). 



Secondly, if an identity exist between two formulas which are 

 partly integer, partly proper algebraic fractions (of which the numerators 

 are of lower dimensions than the denominators) the integer and fractional 

 parts are separately equal. 



To demonstrate this principle, let 



represent such an identity, where each symbol denotes an entire function 

 of X, and the dimensions of P, P' are respectively lower than those of 

 Q, Q'; then we have 



(N-N)QQ = PQ- PQ'. 



If therefore N—jV' be not identically nothing, we shall have the 

 entire function, represented by the left-hand member, identical with one 

 of lower dimensions ; but this is impossible, because in integer formulae 

 we may equate like powers of x, hence we must have iV=iV' and, 

 therefore also, 



Z! - ^ 



Q- Q' 



By means of this principle, we shall be able to expand a given 

 entire function P, in terms of other given functions, whenever such an 

 expansion is possible. 



