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



This theorem is arithmetically true, and may be verified 

 approximately by calculation, when 71 is greater than unity ; 

 but for other values of w, whether positive or negative, we do 

 not approximate to arithmetical truth by taking additional 

 terms of the series. If we multiply together a definite number 

 of factors of the form 



(-D"'(-D"'0-D''-0-.-)"'. 



we obtain the sum of the reciprocals of consecutive ordinals 

 up to the next prime beyond p exclusive : and besides this, 

 we obtain the reciprocals of an infinite number of scattered 

 ordinals, whose aggregate sum does not diminish without limit 

 as p increases. This may be expressed by saying, that in the 

 equation 



the two infinities are not the same, and the difference between 

 them is a material consideration. 



The proposition above given readily conducts us to expres- 

 sions for Bernoulli's numbers in terms of prime numbers; 

 which, however, we reserve for future discussion. 



(\v \ X 



— denote the whole number — when x is di- 



visible by />, and the next whole number below — when x is 



not divisible by p, then the continued product 1 .2.3.4.... ;r 

 or [.r] may be expressed by means of its prime factors in the 

 form 



2G)<:0<5)-3G)<?)<?)-- /tXjXP- ... 



all the primes up to x being employed. 



This theorem may be deduced from the considerations put 

 forward in the first proposition ; but as I find that it is de- 

 monstrated by Legendre (Introduction, p. 10), I omit the 

 proof. 



The familiar association of the arithmetical continued pro- 

 duct \_x\ with its algebraical extension r(a; + l), where Vx is 

 a solution of the functional equation (^{x-{-\)=x<^Xi conducts 

 us naturally to the investigation of an algebraical extension 

 for this continued product when resolved into its prime fac- 

 tors; an investigation which throws us back on the funda- 

 mental principles upon which algebraical extensions of this 



