72 REPORT—1896. 
= Bom41 
(2m + 1) (2m+ 2) 
and the series will be semi-convergent, if 7 >2, as it always is in statistical 
problems. Throughout m is tobesummed for all integer values from 0 to 
co, and the logarithms are to Napier’s base. 
The results (iii.) and (iv.) allow us to calculate F (7,1) and G (7, v) to 
any degree of accuracy that may be required. If we stop at the m term 
in x (7, #) then the error in the value of y (7, ¢) will be less than 
where X2m+1 (¢) {(5)?" +? — cos?” +1 @ cos (2m+1) 9}, 
(—1)n X2m41 (o) 
(fr)2m+1 j 
Now, it is easy to show that although y2»41() has several maxima 
given by 
sr 
®= 3 (m+) 
where s is an integer, still its absolutely greatest numerical value is given 
by ¢=0, and it is then equal to 
Bom4i 
ee (ante 
2m +1) (2m+2) (I=) ) 
Thus, if we stop the calculation of x (7, ”) at the m term, we shall not: 
make an error + or — in its value so great as 
Bom+1 ee eel 
(2m+1)(2m+ 2) (dr)2m+1 
We accordingly obtain the following system of the maximum errors 
possible when we stop at successive terms in x (7,7): 
Term stopped at : Ist 2nd 3rd 
Error less than: + °0625000/(4r) +-0026042/(4r)? +-0007812/(47)°. 
Term stopped at : 4th 5th 6th 
Error less than : +£-0005929/(3r)7_ +-0008409/(3r)9 +-0019171/(4r)". 
Term stopped at : 7th 8th 9th 
Error less than :. +-0064099/(47r)!3 +-0295499/(4r)! +-1796437/(4r)"7, 
Term stopped at : 10th 
Error less than: +1-3933926 /(4r)!9. 
Now, if r=2, we ought to stop at x; to get the closest result from our 
semi-convergent series. We shall then make an error of less than 6 in 
the 10,000. Such a result is generally close enough for statistical 
practice, but is hardly sufficient for the purposes of pure mathematics. 
However, if we start with r=4, and proceed only to the fourth term, 
X7 we should obtain results only showing error in the sixth place of 
decimals. If we calculate y (7, v) up to yo, we have an error less than 
‘000002 for r=4, and less errors for larger values of 7. Finally, if we 
limit ourselves to values of r= or < 6, we shall find that by proceeding to y, 
only we have errors of. less than -0000003 in our results. As the tables 
of logarithms and trigonometrical functions in general use do not go 
beyond seven figures, it does not seem necessary for practical purposes to 
go beyond x, in the calculation of the y-functions. If we, then, start 
our tables with r=6, we shall obtain results for x (7, ») certainly correct to 
