470 



in 



= e" 



HEPOET — 1878. 



e or . e a{>J(x-+xh)-x) 



[l + a { V (x 2 + xh) - x} + J^ { V (x* + xh) - x }* + &c, 



and in 



— ax paW(x*+x?i)+x') 



e •* 



— ax | 



= e 



[l + a { VC.r 2 + «») + x J + J?i { vV 2 + a-A) + x }* + &c, 

 we obtain by means of the expansions 

 ,/(.r» + .rA)-.,t w = j-A«]l-nJ.^"C w + 3) * a _ "(« + 4) (n + 5 ) A» + &c _) 



4a- 1.2 4 2 .r 2 



1.2.3 4 3 .i' a 



h/r,^^ + ,4'^^li-^A^'(^-3).^ «(«-4) («-6 ) A» | 



I ' ) { 4a- 1.2 -4V 1.2.3 4li 3 | 



the general integrals of the differential equation in the different forms of which it is 

 known to be susceptible. 



15. On Certain Special Enumerations of Primes. By J. W. L. Glaishek,, 



M.A., F.B.8. 



The paper related to (1) prime-pairs, and (2) primes of the form 4m + land 

 the form 4m + 8 enumerated separately. 



I. Prime-pairs. — By a prime-pair is meant a pair of primes separated by only 

 one number ; thus, 11 and 13, 17 and 19, 29 and 31, &c, are prime-pairs. It is 

 clear that as the number of primes decreases as we ascend higher in the series of 

 numerals, the number of prime-pairs must decrease also, and the object of the 

 enumeration was to examine 'u rapidity of this decrease. The enumerations 

 relate to the first hundred chiliads (100,000 numbers) of each of the six millions 

 over which Burckhardt's and Dase's t ables extend ; and the number of prime-pairs 

 in each ten chiliads is shown in the following table : — 



f 6 089 999 T 

 * The prime-pair - g'oon'ooi f' is counte ^ ^ belonging to the group 6,080,000 — 



6,090,000 only. 



The explanation of the table is that the number of prime-pairs between and 

 10,000 is 206 ; between 10,000 and 20,000 is 137 ; . . . between 90,000 and 100,000 

 is 108 ; between 1,000,000 and 1,010,000 is 84 ; between 1,010,000 and 1,020,000 



