concerning Numbers. 47 



quence of the computation being made from the commence- 

 ment of the series, where the variation of the logarithm is 

 greatest. 



With a view of applying the formula in somewhat higher 

 parts of the series, I have computed the primes from 2,010,000 

 to 2,019,000 at 620, while Burckhardt's Tables give 617 ; and 

 from 2,982,000 to 3,0 1 8,000 at 2,4 1 3, which agrees exactly with 

 the Tables ; but these ransfes are too limited to enable us to 

 judge of the formula. 



Since the preceding table was computed, I have found the 

 following formula, which gives the most accurate and expedi- 

 tious method of calculating the logarithm-integrals of large 

 numbers : 



lix'-Iia?= (a?'-^)t;-^^V{ D -/t;D + 2fVD2- . . } (^-^ V 



where D denotes differentiation with regard to t. 

 When ^=1, this becomes 



{3t^-'X)v-xv^{\-v{s-2)-\-v\6-'2s)-'iP{9s-~2^) 

 + V*(120— 4.4s)— 1;^(265£ — 720) + ..}; 



the general term of the part within brackets abstracted from 

 its sign being 



±i^-'(1.2.3..w)|l-/l -- + — -..+ — ^ \\ 



^ \ ^ 2^2.3 -2.3..njr 



or 



\n+l (w+l)(w + 2)"'~ (w+l)(w + 2)(n + 3) •*'/* 



which diminishes as n increases, and is always positive. In 

 the form 



V ™4-, T «. 8'»+i-e'» £"»/ -7182818 -5634.364 



m m^\ m nr 



_ -4645365 -3955996 _ \ 

 w^ m'^ "/ 



it can be calculated with great ease, particularly when m is an , 

 integer. The following are instances : 



Primes from 300000 to 815484; computed 39089 ; counted 39082 

 1839.39 to 500000; ... 24890; ... 24883 

 331091 to 900000; ... 42819; ... 42778*. 



The circumstance that log^ represents the average distance 

 between two primes at the point x in the ordinal series, gives 



* I subjoin the following Table of logarithm-integrals, which will serve 



