TRANSACTIONS OF THE SECTIONS. 21 



collecting together all the coefficients of :_, 



a; 



and therefore 





h 



li(w+A;)-liy=i ^1 — r^ — 



J-.0-:)l' .hfiHll 



Gog, 2/)' 



an extremelj'^ convenient formula for calculating from Hargreave's principles the 

 approximate number of primes between limits. The last-%^iitten formula was, of 

 course, deduced from the jjrevious one by taking e^'—y and e'''''^=y-'t-Ti. Li the 

 Tables Z;=50,000, and for 2/=l,000,000 the value of the second term only amounted 

 to 6-2, and the third 0-2 ; for y = 1,950,000 the second term was 3-2, and the third 

 insensible, while for j/ =8,000,000 the second term was only O-C ; so that the first 

 two terms were practically sufficient for the second million, with the interval of 

 50,000, and the first alone for the ninth million. It is impossible, in a brief abstract 



like the present, to notice the aOTeement with Lee:endi'e's formula 



^ ° ^ log a^-1 -08.360 



&e. ; but the author hopes to publish the values for the other millions elsewhere. 



The results given in the two Tables above were calculated or counted in duplicate 

 throughout ; and it is believed that none of the values of li.i'— lia; will be found 

 wrong by so much as a unit, though an error of this amount is just possible. In 

 the total, which was formed merel}' by adding the numbers in the Jia;'— li* column, 

 of course a somewhat greater error is possible bj' accumulation. It may be conve- 

 nient here to state that li(l,000,OOOj=Ei(13-8i551)=78627-2 ; li(l,0o0,000) 



=Ei (13-86430) = 82239-9. . . . ; li (1,100,000) = £i (13-91082) = 85840-2. . . . ; 

 li (1,1-50,000) = Ei (13-95527) = 89428-7. . . . These values were not obtained from 

 the first by means of the above Table, but were each calculated independently from 

 the semiconvergent series 



Eia; = e'' 



j^ ij-LJ 1 • 2 • 3 1.2.3.4 1 



Hargreave has given a formula which is no doubt a particular case of that in this 



Eaper (though I have not yet compared them) ; but either some of his constants must 

 ave been erroneous, or he must have made errors]of calculation, as all the numbers 

 given in the Table on page 48 of the ' Philosophical Magazine ' for July 1849, which 

 was calculated by means of it, seem to be more or less inaccurate (see ' Philoso- 

 phical Transactions,' 1870, p. 386 ; the arguments are identical, as, in fact, HargTeave 

 has taken integer arguments of the exponential integral, viz. Ei 1, Ei2, &c.). 



It maybe added that the number of primes between 1 ,000,000 and 2,000,000 

 was computed by Hargreave and found to be 70,430, which differs by only a imit 

 from the value in this paper (which value, as before remarked, could very well 

 have been inaccurate by even more than this amount) ; and this completely verifies 

 the accuracy of the numbers in the li x'—\\ x column of the first Table between 

 two and three millions. Hargreave (Philosophical Magazine, August 1854) found 

 the number of " Primes counted" up to one million and between two and three 

 millions to be 78,403 and 67,751 respectively ; while the formula gave 78,626 and 

 67,916, the discrepancies being much greater than that which is here foimd for the 

 second million, where the ditterence was only 9. The numbers I have found for 

 the " Primes counted " differ from Hargreave's ; but as they have as yet been 

 counted but once, no great reliance can be placed on them. The formula values I 

 have not yet calculated. 



On a Verijication of the Probability Function. 

 By J. E. HiLGARD, U. S. Coast Survey, 



