74 REPORT—1896. 
and 21°. The values of x might be taken at once from our table, but the 
method of calculation is illustrated by calculating them ab initio. The 
following are the logarithmic values of the y’s to base 10 :— 
log x, =2-9208188 + log (:250000— cos ¢ cos @) 
log x3=3°4436975 + log (062500 — cos * cos 3) 
log x;= 48996294 + log (:015625—cos *@ cos 59) 
log y;=4'7746907 + log (:003906,(25)—cos “¢ cos 7). 
These are obtained by inserting the values of the Bernoulli numbers.! In 
the case of the trigonometrical quantity in the argument of the last 
logarithm being greater than the numerical constant, care must be taken 
to make the corresponding x negative. 
We find 
= 20° o=21° = 20° 35!  =20° 35’ 
By [ater- Direct 
polation Calculation 
xX, — 052752 — 051798 —°052196 —-052200 
x3 —000979 —-000852 —-000905 —-000905 
x, +°000113 +-000158 +:000139 +-000140 
X7 +°000297 + 000311 +:000305 -+-000306 
x (9, 20°9)=— 0117119 x (9, 21°) =—-0115013 x (9°35, 20° 35’) 
x (10, 20°)=—-0105425 x (10, 21°) =—-0103527 ~ = —-011157 
F (9, 20°)=1°374821 (9, 21°) = 1-456858 
F (10, 20°)=1-394909  F(10, 21°) = 1-488643 F (9-35, 20° 35’) 
F (9°35, 20° 35/’) =1-429911 
35 is oD by direct 
265 91)° “N29 “9 y airec 
=F (9, 20°) +5" (-082037) + = (-020088) Pa pecs 4 
=1:429707 by interpolation. 
Thus we see that if tables of F (7, »), proceeding by units and degrees, 
are calculated, the value of F (9°35, 20° 35’), as found by interpolation 
from the tables or direct calculation, would only differ by two units in the 
fifth place of figures. Such a degree of approximation is more than 
sufficient for practical purposes in statistics. Had we used values of the 
x’s correct to the seventh place of figures and used second differences, our 
results would have agreed to the sixth place of figures. Should this not 
suffice for the more exact purposes of pure mathematics, our table would 
still serve as a skeleton to be filled in at smaller intervals of the 
variables, when necessity arises. 
So far as the value of F (7, ¢) we have selected is concerned, x; and x; 
contribute no sensible portion up to the sixth place of decimals. They 
have been included above, however, to indicate how their values for 
1 Higher values of x are given by 
log x= 4°9251836 + log (-000976(56) — cos ° cos 99) 
log x), = 3°2827414 + log (-000244(14) —cos "9 cos 11) 
log x), = 3°8068754 + log (:000061(04) — cos '¥p cos 139) 
log x,, = 2°4705670 + log (-000015(26)—cos'°9 cos 15¢) 
log x,,= 1:2544136 + log (-000003(82) — cos "p cos 17) 
log X,,= °1440741+ log (-000000(96) — cos *p cos 19). 
Still higher values of x may be found almost exactly from 
2 2n 
Xan+1= — (Bx jansa 08 2n+1h cos (2n+ 1). 
