1878.] Mr Glaisher, On circulating decimals. 197 



portion from 1 to 80,000 is printed in the Proceedings of the 

 Royal Society, vol. xxii. pp. 200—210 and 384—388, and the 

 remainder is preserved in the archives of the Society (see Proc. 

 Roy. Soc. vol. xxiii. p. 260 and vol. xxiv. p, 892). In the forma- 

 tion of these tables, the actual periods were not of course found, 

 but the numbers of digits in the periods were obtained by different 

 processes ; the object of these being to determine when the 

 remainder unity lirst occurs, it being known that this must corre- 

 spond to a number of quotient digits equal to ^ — 1 or to a sub- 

 multiple of p — 1, where p denotes the prime. Some account of 

 the methods of finding the number of digits in the period of a 

 prime is given in papers by Mr Shanks and Professor Salmon in 

 the Messenger of Mathematics, new s.eries, vol. ii. pp. 41 — 43, 49 — 

 51, 80. 



In 1868 Mr G. Suffiekl of Clare College, and Mr J. K. Lunn of 

 St John's College, Cambridge, published a folding sheet ^ coutain- 

 iug the complete period of the prime 7,699 which consists of 7,698 

 digits, and the process by which the number of digits in the period 

 of this prime was determined beforehand is explained in detail. 



Taking this prime as an example, the number of digits in the 

 period must be either 2, 8, 6, 1288, 2566, 3849 or 7698, and the 

 process of finding which of these is the true number consists in 

 determining whether each of the remainders, after 2, 3, 6,... quo- 

 tient digits have been obtained, is or is not unity. In any case 

 the remainder, after 7698 digits have been obtained, must be 

 unity, and the question is whether a remainder unity presents itself 

 at one of the earlier stages at which it may appear. Through an 

 error of calculation too small a value might be assigned to the 

 number of digits in the period (as e.g. 3849 instead of 7698 in this 

 instance); but the converse mistake of obtaining for the number of 

 digits a multiple of the true number (when this number is a sub- 

 multiple of p — 1) is much more likely to occur. For supposing the 

 period to contain a digits, where na =]3 — 1 ; then if the remainder 

 after a quotient digits be (wrongly) found not to be unity, but after 

 ma (in being a submultiple of n) quotient digits be (rightly) found 

 to be unity, the number of digits in the period would be assigned 

 as ma. It is for this reason that some of the processes employed may 

 be unsatisfactory, and that it is not safe to rely upon the complete 

 accuracy of a table of this kind unless confirmed by another table 

 independently calculated. Five years ago I had a comparison 

 made between the tables mentioned at the beginning of this 

 section, and the discrepancies were numerous : a good many of 



1 This sheet, which is dated April 29th, 1863, seems to have been published for 

 private circulation : I was never able to see a copy till Mr W. H. H. Hudson kindly 

 lent me his. 



