252 



where A,. — -; 



REPORTS ON THE STATE OF SCIENCE, ETC. 

 1 



3' 4:'- 



ad inf.. 



but for general purposes the best formula seems to be 



9W=Y+|log, (x^+x)+:R 



where y=0-57721 56649 01532 86060 . . . 



(4) 



and 4=6 

 R 



1 IQ 1^ 



^ '5 525 ' 375^ ' 



the first two terms of which are suiScient for tabulation purposes to seven or more 

 decimal places when x exceeds seven. 



The expression for R may also be written in either of the following forms : 



1 1,4 1 , 32 , 



,1 • • • 



R = 



6(a:2 



orR = 



1 



30(x^+x)'^ 315(x2+xf 105(a;2+a;) 

 19 43 



; + 



2310(a:-+x)5 



6(a;2+a;+^)^3150(a;'i+xf 6250(a:-2+x)4" 



-6{x^+x+l) 



+ li75+3l50;^"'+")~ 



' U2 



125^6250 



(x^+x)-i+ 



The method adopted for the tabulation was the direct calculation and addition of 

 the reciprocals of the successive integers up to a;=61, and the calculation by formulae 

 (2)and(l) for the half integers, and the use of (4) for x =50-1 to 50-9 to 16 decimals, 

 from which by means of (1) all the other values of the function were tabulated. The 

 last place of decimals is necessarily approximate, but I do not think it is anywhere 

 more than a unit out. 



The later part of the work was much facilitated by using a wonderful table of 

 reciprocals published in 1823, and kindly presented to me by the Rev. J. J. Milne. 



A further check was obtained by independent calculation by (4) of the part 

 between a;=10-l and x=10-9. Also ^(O-l) was calculated from (3) and found to agree 

 weU with the value already calculated. 



The table could be extended beyond its present limits by means of (1), or it could 

 be dispensed with by using the formula (4). 



Interpolations in the present tables could be made by either of the formula given 

 below, or by the 'difference' method, which would, however, be very cumbrous in 

 some of the earlier portions. 



With a view to filling in a more complete table I have tabulated the function 

 between a;=50 and 51 to 001 intervals to 10 decimals. This would be a foundation 

 for more detailed tabulation in the other dekads. 



In an Appendix I give the method of obtaining the important formula (4), and 

 also allied formulae from which the sums of reciprocals of other odd powers could be 

 obtained. 



Interpolation Formula. 



First Method ; <p(a+x) = (l + A)-r(p(a) leads to the series 



<p(a+a:)=q)(a)+-— Y + 



x(\—x) 



x(l— x)(2— a;) 



a+1 2(a+l)(a+2)^3(a+l)(a+2)(a+3) 



Second Method : 9(a)=y+i log a(a+l)+Eo 



cp(a+a;)=Y+-J log (a+x)(a+l+a;)+Ri 



+ .. 



/ , ^ / > ill «+« , , a+l + a;] 

 <p(«+ x;- 9(a) = ^ log ^ + log ^^^ [ 



='( 



+ 



1 



c/ 3 



1 



which is practical]}' equal to 



2a+x 2a+2+a;/ ' 3" 



X 



-+i 



2a+x 2a+2+a; 

 when X is small compared with a. 



l(2a+xy 

 (Ro 



.+ 



-(Ro-Ri) 

 1 



(2a.+ 2+xf 



Ri) 



1 



+ . .-(R,-Ri), 



