IS 



42 Mr. C. J. Hargreave's Analytical Researches 



and suppose ourselves not to have any previous acquaintance 

 with their properties further than necessarily flows from the 

 circumstance that they are the elements of which the natural 

 numbers are composed. We have then by virtue of this fact 

 the equation 



where Py is the last prime in the series of ordinals up to ^ ; 

 and the question suggests itself, what meaning is to be attached 

 to the second side of this identity, when the first side is used, 

 not in the arithmetical sense, but in the extended sense which 

 is ordinarily understood by the term r(,r+ 1). What can we 



propose as a proper analytical equivalent of ( — j. . ? To th 



question it would perhaps be difficult to give a general satis- 

 factory answer; it will be sufficient for the present purpose to 



express the analytical equivalent of ( ) ~ I — ) ' ^"^ *^ 



is submitted that if any such equivalent exists, it can only be 



— in the ordinary sense. In this investigation x and p^ are 



P\ 



symbols to which we do not assign a definite arithmetical 



value ; we know merely that j9j is a prime. Now ( — ) niay 



differ from the ordinary fraction — by any one of the quantities 



12 3 jOi-2 Pi-1 , a; + Aar . ,., 



0, — , — , — , ...-^-i , ^-^ ; and may m like man- 



Pi Pi P^ Pi Pi Pi 

 ner differ from the ordinary fraction by any one of the 



same quantities ; and if we are at liberty to adopt in its sim- 

 plest form the principle before alluded to, — that the analytical 

 equivalent of an indeterminate expression is the arithmetical 

 mean of all its possible values, — we shall have no difficulty in 

 concluding that the proper expression for the difference in 



A .7.' ^x 



question is — ; for its value may differ from — by any one 



of a set of quantities the arithmetical mean of which is zero. 



It will be observed that in the arithmetical identity of which 

 we are speaking, we are at liberty to consider the series 





+ 



as extending ad hifinitum^ without thereby impairing the truth 

 of the equation, or detracting from its purely arithmetical 



