146 THE LAWS OF 



of V by giving to i and t the various values of which these 

 numbers are susceptible, and taking the sura of all the parts 

 corresponding to the different terms in the expansion of the func- 

 tion f(x' 9 y, z'). 



Although in the present Article we have hitherto supposed f 

 to be the characteristic of a rational and entire function, the 

 same process will evidently be applicable, provided /(#', y' 9 z) 

 can be expanded in an infinite series of the entire powers of 

 cc', y ', z and the various products of these powers. In the latter 

 case we have as before the development 



f(x', y', z') =/"! +/" 1 +/ +/"" + &c. 



of which any term, as for example f' (i \ may be farther expanded 

 as follows, 



/*" + &c. 



and as we have already determined V t (i) or the part of V cor- 

 responding to f{ (i >r i+ ', we immediately deduce as before the 

 required value of V, the only difference is, that the numbers i 

 and t, instead of being as in the former case confined within 

 certain limits, may here become indefinitely great. 



In the foregoing expression (11) 13 may be taken at will, but 

 if we assign to it such a value that - may be a whole num- 



ber, the series contained therein will terminate of itself, and 

 consequently the value of V t (i) will be exhibited in a finite form, 

 capable by what has been shown at the beginning of the present 

 Article of being converted into a rational and entire function of 

 x, y, z, the rectangular co-ordinates of p. It is moreover evi- 

 dent, -that the complete value of V being composed of a finite 

 number of terms of the form V t (i} will possess the same property, 

 provided the function /(a/, #', z} is rational and entire, which 

 agrees with what has been already proved in the second Article 

 by a very different method. 



5. We have before remarked (Art. 2), that in the particular 



*M n_-ii A. 



case where fi = , the arbitrary constants contained in 



