128 ON THE LAWS OP 



and as in this equation, the function on the left side contains 

 none but the even powers of the indeterminate quantity r, whilst 

 that on the right does not contain any of the even powers of r, it 

 is clear that each of its sides ought to be equated separately to 

 zero. In this way the left side gives 



F=r;+2y+7y + r;+&c (5). 



Hitherto the value of the exponent /3 has remained quite 

 arbitrary, but the known properties of the function T will enable 

 us so to determine /3, that the series just given shall contain 

 a finite number of terms only. We shall thus greatly simplify 

 the value of F, and reduce it in fact to a rational and entire 

 function of r 2 . 



For this purpose, we may remark that 



r (0) = oo , r (- 1) = oo , r (- 2) = oo , m mfinitum. 



If therefore we make h /3 = any whole number positive 



or negative, the denominator of the function (4) will become infi- 

 nite, and consequently the function itself will vanish when s is 



7? 



so great that -+ /3 + t + 3 s' is equal to zero or any negative 

 2i 



number, and as the value of t never exceeds a certain num- 

 ber, seeing that f(r' 2 ) is a rational and entire function, it is 

 clear that the series (4) will terminate of itself, and V become a 

 rational and entire function of r*. 



2. The method that has been employed in the preceding 

 article, where the function by which the density is expressed is 

 of the particular form 



p-(l-OW), 



may, by means of a very slight modification, be applied to the far 

 more general value 



p = (1 - r' 2 ) "/(*', y', ) = (1 - a? -y" - ")'/V, *', <0, 



where / is the characteristic of any rational and entire function 

 whatever: and the same value of /3 which reduces F to a 

 rational and entire function of r 2 in the first case, reduces it in 



