316 



REPORTS ON THE STATE OF SCIENCE, ETC. 



Sine and Cosine Integrals. Si (a;) and Ci (a;). 



These mtegrals were tabulated ^to ten places of decimals over the range »;= 5-0 to 

 a; = 20-0 by 0- 1 intervals and published in last year's Report. Values of these functions 

 for the range a;=20-0 to 40-0 by 0-2 intervals were required in the construction of 



tables of Bessel function derivatives -=^ - Jv (x) when v is half an odd integer. For 



large values of x, the asymptotic series were used in the manner set out in the 

 prefatory note to the tables published last year. Twenty values were computed in 

 this way for integer values of x from 20 to 40. The differential coefficients of these 



were next computed for smaller intervals and differenced, and 



sm X 



cos X 



functions '— and 



X X 



the iirst difference of Si {x) and Ci (x) obtained from the central difference interpola- 

 tion formula 



A, = ii.fi -i u.S-fi+^~u.8*h - ^-^ fxS8/i+ . . . 

 "1 W4 i2^"-'*^720'^ •'* 60480^ ■'* 



, X i -ii. sm X cos X 

 where / represents either or • 



XX 



The comparison of the difference of two entries of Si (x) or Ci (x) for integer values 

 of X and the sum of the first differences as calculated above served as a check on the 

 work. An error in the tables of Si {x) published last year has been discovered and ia 

 here corrected 



Si (5-3) = +1-49731 50636 



A short table of Si {x) to five decimal places for integer values of x from 16 to 60 

 appeared in 1914 in a paper by Lord Rayleigh. i Bretschneider also has tabulated - 

 the sine and cosine integrals for a;=0-0 to 1*0 by 0-01 intervals and a;=l-0 to 7-5 by 

 0-1 intervals to ten places. 



1 Lord Rayleigh. Proc. Roy. Soc, vol. xc, p. 320. (1914.) 



2 Bretschneider. Zeit. fur Math. u. Phys., Band vi, 127-139. 



(1861.) 



