ON CALCULATION OF MATHEMATICAL TABLES. 275 
Calculation of Mathematical Tables. — Report of Committee 
(Professor J. W. Nicnotson, Chairman ; Dr. J. R. Atrey, Secretary ; 
Dr. D. Wrincu-Nicnotson, Mr. T. W. CHaunpy, Professors L. N. G. 
Fiton, E. W. Hopson, Mr. G. Kennepy, and Professors A. Lopes, 
A.E.H. Love, H. M. Macponatp, G. N. Watson, and A. G. WexstER). 
As indicated in last year’s Report, the following tables have been prepared for publi- 
cation for the Toronto Meeting :— 
(1) Tables of sin 0 and cos 0 to fifteen places of decimals for values of § in radians 
from 10-0 to 20-0 by intervals of 0-1 and from 20-0 to 50-0 by intervals of 0-5 radian, 
supplementing tables of these functions published in earlier reports of the Committee, 
viz., sin 8 and cos @ to eleven places of decimals for values of § from 0-000 to 1-600 by 
intervals of 0-001, with a short subsidiary table for purposes of interpolation, and Dr. 
Doodson’s table giving one hundred values of sin 6 and cos 6 to fifteen places, 0 ranging 
from 0:1 to 10-0 radians (Report, 1916); also sin 6 and cos 6 calculated to twenty- 
four places but reduced to fifteen to be uniform with the previous table for 0=0 to 
100 radians (Report 1923). 
(2) Tables of the Lommel-Weber functions (),(x) andQ,(2) were calculated seven 
years ago with a view to their publication in the report of the Committee for 1917, but 
war conditions made this impossible. These functions are related to those of Struve 
which Professor Watson has recently given to seven places of decimals in the collection 
of mathematical tables in his ‘ Theory of Bessel functions.’ 
(3) Tables of the Bessel-Clifford* functions C,(x) and C,(x) which were computed 
from the relation C,(x)=J,(2 Vx) and C,(x)= ae These functions of zero and 
unit orders are of some interest, apart from their practical applications, and the con- 
struction of the tables did not present much difficulty. For functions of fractional 
order, however, the calculation from the well-known Bessel functions is troublesome 
and perhaps unnecessary. . 
For next year’s Report, it is suggested that the Association might undertake the 
publication of tables of the Bessel functions J4n+1 (x) to twelve places of decimals 
2 
for z=1 to x=20 : for positive orders, the calculations have been carried to the point 
where the first significant figure is in the thirteenth place, and for negative orders 
where the value of the function does not exceed unity: also tables of the Lommel- 
Weber functions, Q4(x) and Q_}(«) to six places for values of the argument x from 0-00 
to 20-00 by intervals of 0-02, and similar tables of J;(x) and J_}(”). The computation 
of tables of the Confluent Hypergeometric function} M(«. y. x), in particular for y=}, 
is nearing completion. Tables of this kind have been found especially useful for the 
numerical solution of differential equations of the type 
dy dy = 
gaz + (PET) » 7 + (la*-+-ma+-n)y=0 
and other equations occurring in physical and engineering problems. 
Sines and Cosines (0 in radians). 
From the tables of sin 6 and cos @ published in last year’s Report and the known 
values of sin « and cos «, where «—0-1 radian and 0-5 radian, 
sin 0-1=-+0-09983 34166 468281 : 
cos 0-1=+-0-99500 41652 780257: 
sin 0-5=-+0-47942 55386 042030 
cos 0:5=+-0:87758 25618 903727, 
the following table was constructed for intermediate values of 0 by the relations 
sin (0-+-«)=sin 6 cosa+-cos@ sin «, etc. For this purpose, the first hundred multiples 
* Sir George Greenhill, Phil. Mag., vol. 38, November 1919. 
t Phil. Mag., vol. 36, July 1918. 
