ON CALCULATION OF THE G (1, v)-INTEGRALS. 73 
the sixth figure. The earlier portion of the table may then be calculated 
from the formula of reduction : 
Bop cc Us 2 ig aie , 
CCl carircays Thee z eee) 
and the entire tables will then be correct to the sixth place. 
3. It may be observed that the formula (iv.) is of considerable signifi- 
cance. It is quite independent of the nature of 7, whether fractional or 
integer, and thus shows that there is no abrupt change in the value of 
G (r,») when we pass from integer to fractional values. 1t thus justifies 
interpolation between integer values of vr, in order to find the value of 
the function for x fractional. It might be supposed, if for statistical 
purposes it is sufticiently accurate to interpolate between integer values 
of r, and as G(r, v) is directly integrable in a terminable series ' when 
is integer, that to use this latter series would be the readiest means of 
calculating tables of G(r,v). But this is far from being the case, and for 
the following reasons : 
(i.) We have always as many terms to calculate as in finding x (7, 9), 
and often many more. 
(ii.) These terms are not the same for all values of ¢, and must be 
calculated afresh for each pair of values of ¢ and 7; 2.¢., they cannot be 
broken up into ¢-factors and r-factors, and the former and latter calculated 
independently and once for all. 
Hence, even when 7 is an integer the calculation of G (7, 1) proceeds 
best by aid of the x-functions. 
4. The process of calculation has accordingly been the following :— 
(a) The calculation of a table of y-functions from x, to x; for values 
of » from 0° to 90°. This table will be found at the end of this paper, and, 
until the complete tables of F (7, ) are ready, will enable the value of 
F (r, v) for any value of r and » to be found with a fairly small amount 
of labour. 
(6) Very considerable progress has been made with the calculation of 
F(r, v) from the x-functions for selected values of 7. It is proposed to 
fill in the gaps by means of the reduction formula (v.). A test of the 
accuracy of the calculations will thus be obtained by the agreement of the 
directly calculated values with those obtained by reduction from the last 
directly calculated value. 
The arithmetic has proved much more laborious than was at all 
anticipated at the start. It was originally undertaken by Mr. H. J. 
Harris, assistant to Professor M. J. M. Hill at University College, 
London, but the whole of the calculations have been again and indepen- 
dently worked out by members of the Department of Applied Mathemat‘cs 
in that College. 
5. It seems desirable to illustrate the method of calculation, and to 
show, in one case at any rate, the degree of accuracy obtainable by inter- 
polating between integer values of 7 and values of ¢ proceeding by 
degrees. 
Let it be required to calculate F(r, v), when r=9°35 and r=3-51133. 
Tt will be found that ¢=20° 35’, and hence, when the tables are com- 
pleted, it will be necessary to interpolate between r=9 and 10 and ¢=20° 
1 By expressing sin "@ in cosines or sines of multiple angles. 
