ON CALCULATION OF MATHEMATICAL TABLES. 287 
iculation of Mathematical Tables.—feport of Committee 
(Professor J. W. Nicuouson, Chairman; Dr. J. R. Atrey, Secre- 
tary; Mr. T. W. Cuaunpy, Professor L. N. G. Fron, Colonel 
Hrertsuey, Professor E. W Hosson, Mr. G. Kennepy, and 
Professors A. Lopasr, A. E. H. Love, H. M. Macponatp, G. N. 
Watson, and A. G. WEBSTER). 
In the Report for 1922 reference was made to mathematical tables calculated for the 
_ Association without the assistance of grant from the Committee. The publication of 
these tables was deferred ; the Report for the present year includes in Part I. the 
tables of sin 9 and cos 0 for 0 in circular measure from 1 to 100 radians, supplementing 
: those computed by Dr. Doodson in the 1916 Report, viz. sin 0 and cos 0 to fifteen 
places of decimals for 9 = 0 to 10 by intervals of 0-1 radian. 
Tables of Bessel and Neumann functions, where the order and argument are equal 
_ or differ by unity, have been calculated to six places of decimals and published in the 
Report for 1916, the order and argument having integer values only. In Part II. 
will be found tables of these and other functions, the integrals of Schliifli and the 
_ Lommel-Weber functions, where the order and argument are not restricted in value, 
but contain both integral and fractional values, the order of the functions ranging 
from 0 to 10 by intervals of 0-25. The work of calculation, especially that of the 
_ preliminary tables required for Part II. of the Report, has been much relieved by the 
use of an arithmometer kindly lent to the Secretary. 
Recently, Prof. A. E. Kennelly has placed at the disposal of the Committee tables— 
_ to six places of decimals—of Bessel functions for a complex variable. These functions 
are equivalent to the classical ber, bei, ber’ and bei’ functions of Kelvin, but are more 
} convenient to use in electrical engineering problems. On account of their practical 
importance, the Committee feel justified in undertaking their publication in Part ITT. 
_ of this Report. 
The other functions referred to in last year’s Report, Bessel-Clifford functions 
-C,(#) and C(x) and Lommel-Weber functions Q,(%) and Q(x), and further tables of 
the sine and cosine functions are reserved for later publication. 
Part I. 
Sines and Cosines (9 in radians). 
The values of the sine and cosine of unit angle (radian) have been calculated* to 
105 places of decimals by Bretschneider. Using only 30 places, the sines and cosines 
of the following angles (radians) were found in succession, 5, 10, 50, 100, the last 
calculation being correct to 25 places of decimals. To obtain the values of these 
functions for intermediate angles, e.g. 20, 30, 40, etc., it was found convenient to 
construct a table of the first hundred multiples of sin 10 and cos 10 to facilitate the 
calculation of the products when sin 10 or cos 10 is a factor. In a similar way, by 
employing a table of multiples of sin 1 and cos 1, the remaining values of sin 0 and cos 0 
were obtained. Hach result was checked by those already computed, sin 54 by sin 52, 
and so on. 
A further check was made by a direct calculation of sin 7] and cos 71. 
Thus 71 (radians) = 22-6x+ 9, where 
; 6 = 0-00000 60288 70672 81074 42595. 
Hence sin § = 0:00000 60288 70672 77422 20829 
and 1— cos § = 0:00000 00000 18173 64079 46837. 
Also sin 71 = sin 22-67% . cos 6+ cos 22-6x . sin 0 
P = cos 18° . cos  —sin 18° . sin 9 
= + 0:95105 46532 54374 63665 6570 
cos 71 = cos 22:6 . cos 8 —sin 22-6r . sin 0 
= —sin 18°. cos 0 —cos 18°. sin 0 
= —0-30902 27281 66070 70291 7494 
verifying the values in this example, and indirectly the whole table, to 24 places of 
decimals. Vifteen places only are given in continuation of Dr. Doodson’s table. 
* C. A. Bretschneider, Archiv der Math. u. Phys., vol. 3, 1843, pp. 28-34. 
1923 x 
