192 Mr. Drach on the Enumeration of Prime Numbers. 



merit of the compound. Whether or not these two com- 

 pounds are isomeric, remains yet to be ascertained. 



The simplest method of preparing the phosphorescent com- 

 pound is to heat together 12 parts of cyanide of mercury, 1^ 

 of boracic acid, and 1 of sulphur. 



The compound of phosphorus and nitrogen (discovered 

 by Rose) probably has similar relations, and may rJtrhaps be 

 studied to advantage in connexion with the above ; an easy 

 method of preparing it is to place some chloro-amidide of 

 mercury in a flask, and add from time to time a portion of 

 phosphorus, keeping up a gentle heat all the time, and agi- 

 tating now and then ; and when the phosphorus ceases to pro- 

 duce any decomposition, raise the temperature nearly to red- 

 ness. 



XXXIII. On the Empirical Law in the Enumeration of Prime 

 Numbers. By S. M. Drach, F.R.A.S. 



EGENDRE gives, in his Thcor. des Nomb., p. 395, the 

 -*- i following approximate theorem for the number y of 

 primes in a given limit x : — 



y = x-*- {hyp. log. a? — 1-08366}. 

 This is successively reducible to 



g% m x -s- 2-95548 j and e* = ('338355 .v)* . 



The former equation shows that the limit, and its ratio to 

 the number of containing primes, are respectively the abscissa 

 and ordinate of a logarithmic curve. According to the learned 

 author of the article Primes, in the Pen. Cyc, anno 1841, the 

 deduction of the constant quantity 1*08366 is as yet unknown. 



As however it = 3*14159, &c, and its powers, in conjunc- 

 tion with a finite and generally simple fraction, represent the 

 sums of so many series of whole numbers, it seemed probable 

 that the aforesaid constant was somewhat connected with it; 



5 /— 

 and in fact — v it = 2*95482 has for its hyp. log. 1*0831904, 

 3 



differing little from the former. This difference is insensible 

 for small values of x, only whole numbers being required, and 

 with the increase of the value of x the constant becomes a pro- 

 portionally less fraction of hyp. log. x. 



The annexed table gives the actual number of primes, P, 

 from 10,000 to a million; the third column is the excess 

 of Legend re's y above P; and the fourth presents the similar 



quantity, 3/ 1 assuming — - Viz as the constant. The total re- 

 sulting errors are for column 3, 191 —68 =123, and for co- 



