18 REPORT 1871. 



to the last figure ; there is therefore no doubt of the accuracy of the result to this 

 extent. 



The value given by Callet, in the introduction to his ' Tables Portatives,' starting 

 with the ninth group of five, is 



. , . 46928 08355 51550 58417 2 . . . ; 

 and these figures should be 



. . . 47093 69995 95749 66967 6. . . . 



The thirtv-nintli convergent to the continued fraction (3) gives a result as accu- 

 rate as that found by siunniing the first ninety terms of the series (1) ; but there 

 would be no great disparity between the absolute nimiber of figures formed in the 

 two calculations. The computation of the convergents was, however, far preferable 

 in point of aiTangement and convenience to the calcidation of the successive terms 

 of a series ; for not only were the divisions in the latter replaced by multiplications, 

 which are far more compact, but the work in the former case ran straight forward 

 and reqim-ed no copjdug of results. There is also another very gi-eat advantage in 

 the continued fraction : the great difficulty of performing a piece of work to a consi- 

 derable number of decimal places is the inconvenience caused by the length of the 

 numbers ; and in the above calcidation we get roughly 2/« figures of the residt 

 without ever having to use a nmuber more than n figures long in the work : thus 

 p.,,, and q^g contain each 67 figm-es, and by dividing them we obtain 138 figm"es 

 of the residt ; this advantage is due to the fact that all the numerators in (3), 

 except the first, are equal to unity. It may be remarked that the final division 

 was the most laborious part of the work ; the calculation of p^^ and ^33 required 

 barely 13,000 figm-es, the division about 18,000. 



We can compai-e the number of decimal places aflbrded by (3) and (1) when n 



is large as follows : — The number of places _ yields* is equal to the greatest in- 

 teger contained in 



= log[i{l. 6. .. (4«-6)}{l . 6. .. (4«-2)}] 



:[^ii.s...(..-:„.]=,..[i!rl..-»{rS^)[.] 



-In. J 1 i n^"+^22>»+.i j n 



(after substituting V27rH«''e-» for r(«-|-l)) 



= log{?!ri.0y"}=2«log^-h(4«-5)log2-logn; 



and the number of places obtained fi.'om n terms of (1) is equal to the greatest 

 integer in 



log r(M) = w log -+1 log 277- i log n ] 



so that the «th convergent to the continued fi'action gives more than twice as many 

 decimal places as n terms of the series. 



On certain Families of Surfaces. By C. W. Meerifield, F.R.8. 



The author had already shown that conical and cylindrical surfaces not only 

 satisfy the general equation of developable sm-faces in ditierentials of the seconii 

 order, 



rt=s^, 



but also that on passing to the diiferential equation of the third order, there are 

 two equal roots in the case of conical surfaces and three equal roots in the case of 

 cylindi'ical surfaces. 



* See Proe. Roy. See. vol. xis. p. 514. 



