68 : REPORT—1899. 
This process was used for values of F (7, v) for r=1 to 7, and the 
values of F (r, v) and H (r, v) as found from the y-functions of the 
tables of the Preliminary Report were compared and found to agree with 
these for r=6 andr=7. For r=6 to r=50 the y-function tables were 
used. The values of g are taken from 0° to 45° proceeding by degrees, 
for no instanee has yet been found in statistical investigations in which 
vis greater than r. Should such cases arise in future, then F (r, v) or 
H (7, v) must be calculated from the x-tables for p=46° to 90° given in 
the Preliminary Report. 
(3) The whole of the arithmetical work (which proved far more 
laborious than was initially anticipated) has been undertaken by Miss 
Alice Lee, B.A., D.Sc., Assistant-Lecturer in Physics in Bedford College, 
London. The arithmetic has been done twice independently, Miss Lee 
having been most kindly assisted in the verification of the tables by Miss 
M. Fry, Miss C. D. Fawcett, B.Sc., Miss E. Bramley-Moore, B.A,, and Miss 
L. Bramley-Moore. To the extent of the methods used we think the accu- 
racy of the tables is guaranteed by the agreement reached by the two sets 
of calculations. But sources of error common to both independent calcu- 
lations have been already referred to, and may be indicated more particu- 
larly here. So far as H (7, v) as obtained from (ix.} is concerned, 2x has 
been worked to 9 figures and is certainly correct to 8 figures. 2, loge was 
then found by actual multiplication. We consider, accordingly, that 
log H (r, v) is correct to 7 figures in all cases, from r=6 tor=50. Any 
inaccuracy of log F (r, v) arises from the extra factors in (xi.). To begin 
with, the factor c*=e'*"* appears as rp tan gloge. was obtained from 
T 
180 
10-figure trigonometrical tables of 1596, and the product rp tan @ log e 
obtained by the Brunsviga. We consider, therefore, that 7? tan @ 
log e, like the previous product 2 y log e, should be correct to at least 10 
places of figures. (7+1) logcos » was found from Vega’s 10-figure 
trigonometrical tables by actual multiplication. It is unlikely, accordingly, 
that there will be an error in this product in the 8th place, and we feel 
fairly certain that the method, when all the factors are added in, cannot 
affect the value of log F (7, 1) to the 7th place. Still, the differences in 
the tabulated values of log F (7, v) are in parts of the table considerable, 
and interpolated values, such as we get in practice, can hardly be con- 
sidered as accurate beyond the 6th figure, or even at certain parts of the 
table beyond the 5th figure. This, of course, is sufficient for statistical 
purposes, but if for physical or mathematical calculations it should be 
needful to have the G (7, v) integrals to a closer value, a table will have 
‘to be constructed for much smaller differences of rand ¢ At present 
the physicist or mathematician must use our H (r, v) integral and find 
(cos ¢)"*! and e”*"* for the actual values of 7 and @ by 10-figure logarith- 
mic tables. He will hardly be sure of being correct to the 6th figure, if 
he uses the usual 7-figure logarithmic tables. 
For values of 7 less than 6, formule (xvi.) and (xvii.) have been used, 
as already noted. The factorial denominators were calculated by aid of 
the Brunsviga and Vega’s 10-figure tables. For the hyperbolic sine and 
cosine tables have been calculated to 14 figures by J. W. L. Glaisher and 
F. Newman, but the differences were so large in the part of the tables 
required, that it seemed safer to recalculate e+” for the special values of 
x n°, using the Brunsviga calculator ; tan ¢ was taken from Rheticus’ 
