TRANSACTIONS OF SECTION A. 



471 



is 69, &c. Thus, for example, the number of prime-pairs between 7,070,000 and 

 7,080,000 is 54 ; and between 8,090,000 and 8,100,000 is 43. 



The total number of prime-pairs between and 100,000 is 1225 ; between 

 1,000,000 and 1,100,000 is 725 ; . . . and between 8,000,000 and 8,100,000 is 518, as 

 shown in the last line of the table ; and it is interesting to compare these with the 

 numbers of primes between the same limits which are respectively 9,593, 7,216, 

 6,874, 6,397, 6,369, 6,250. The numbers of prime-pairs are thus rather less than 

 one-tenth of the numbers of primes in the same intervals. It should be stated that 

 among the prime-pairs 1 and 3, 3 and 5, 5 and 7 are counted (2 being ignored), 

 and that among the primes 1 and 2 are both counted. In the 600 chiliads, the 

 greatest number of prime-pairs contained in any one chiliad is 36 in the first 

 chiliad ; while one chiliad (8,014,000—8,015,000) contained no prime-pair. For 

 a more complete account of the enumeration see ' Messenger of Mathematics,' vol. 

 viii. pp. 28-33 (June, 1878). 



II. Primes of the form in + 1 and in + 3. — This enumeration only extends at 

 present over the first' hundred chiliads of each of the first three millions. The 

 results are shown in the following table : — 



The explanation is the same as in the case of the previous table — viz., the 

 numbers of primes of the forms 4w + 1 and in + 3 between and 10,000 are 610 

 and 619 respectively; between 10,000 and 20,000 are 516 and 517 respectively ; 

 between 1,000,000 and 1,010,000 are 391 and 362 respectively, and so on. The 

 enumeration (which was undertaken at the suggestion of Professor Tcliebychef) 

 is scarcely extensive enough to afford valuable results ; but, owing to the great 

 difference in the properties of the primes of the two forms, the comparison between 

 their frequency of occurrence possesses considerable interest. The numbers given 

 in the table are the result of a duplicate enumeration ; but a third enumeration 

 will be required, in order to render it certain that they are absolutely free from 



error. 



16. Notes on Circulating Decimals. Tiy J. W. L. Glaishek, M.A., F.B.8. 



The author alluded to the advantage, in considering the complete theory of the 

 periods of circulating decimals, of including all the periods corresponding to a given 



