TABLES OF THE G (r, v)-INTEGRALS. 67 
This result we may write : 
Fea Rater SA yevtiitit’. OL oils Qe vite) 
}o¢ (r, v) 
where : H (7, v)=V / tol puro, ; ‘ di Oise) 
Vv 
It is this function H (7, v) which has been chosen for the purposes of 
tabulation. Equation (viii.) shows its statistical importance—it enables 
us, knowing the standard deviation « of the observations—to at once 
determine the frequency of mean values. It will approximate more and 
more to / 27 as the frequency approaches a normal distribution, which it 
does when r is large. Hence the differences of H (7, v) will be small, 
and are likely ta be smooth, when 7 is large, and consequently F (7, v), 
owing to the factor (cos ¢)’*! is not capable of very accurate determina- 
tion. 
The relations between the three functions already mentioned are : 
Ei(r, v)=e7#* G (7, 7) ‘ , : ; oes) 
F (r, v)=e" (cos 9)"** H (7, v) . ; ; ae Xe) 
J/r—1 
Gry) =seP2 WH (75-3) : , ; ‘ sch ul) 
G (7, v)=e%**" (cos 9)? H (7, v) ; h(i?) 
Vr-1 
BiG, y= J/r—le* IBY faji?), b { : raise) } 
(cos )"*? 
H (r, vy)=V/r—1 em G (0) : ; ), Hz) 
(08 gy 
so that any one can be found from either of the others. 
(2) But while H (7, v) is clearly the best function to tabulate when r 
is moderately large, it is not so satisfactory when r is small ; for although 
in that case (cos ¢)’*' may be fairly easily found from the tables in 
ordinary use, so that it might seem that F (r, v) could be accurately 
determined, yet the expression for y (7, ¢) now becomes unsatisfactory. 
As has been shown in the Preliminary Report, § 2, we have to deal 
with a semi-convergent series, and cannot for small values of 7 go beyond 
X73; but this may involve an error as large as 6 in 10,000. Accordingly, 
as the tables only proceed by integers, we have used the following results 
which can, for r= an integer, be deduced by direct integration : 
Dia 
at 
F (2r, v)=2 sinh 3 Ty G22) (4P) 0 Gen Givi.) 
F (2r+1, v)=2 cosh 3 pied exp SEE rap Yee 7 on we 
2" (0? +1) (2 +32). (W? +(2r+1)?) (xvii) 
oO H (7, v) was then deduced from these values of F (7, v) by 
xiv.). 
F2 
