ON CALCULATION OF THE G (7, v)-INTEGRALS. 71 
is the quantity actually required in statistical problems, it is F (7, v), 
which will be tabulated. Interpolation between two values of F (7, 1) 
gives better results for G(r, v) than direct interpolation between two 
values of G(r, v). 
It has been shown by Lipschitz (Cre//e : Bd. 56, S. 20) that the well- 
known expansion in terms of Bernoulli’s numbers for log T (7 +1) still 
holds when is a complex quantity ; the remainder after B2,,_1 is 
- Bom+1 1 - 
Li ! 
ae") (2m+1)(2m+2) n2m+1 Co 
where « and «’ are both less than unity. 
We can accordingly use this expansion to obtain a semi-convergent 
series for F (r, 1). 
log F (7, v)=log 27 —r+ 1 log 2+log I (r+1)—log T (47 +1--$17) 
—log [ (4r+1+4+ $77). 
Let r=2 § cos $, v=2/ sin ¢, and let the [’-function terms be calculated 
separately. isis 
log T (r+1)=log / 27+(r+4) logr—r 
Bom+1 1 
S a m 
staat (2m+1)(2m+4+2) x2m+1’ 
log C (474+1—43ri)=log T (fse— +1) 
=log V 27+(Be- +4) (log B—ip) —e-*# 
Bp 1 , 
4+9(—1)" 2m+1 (2m+1) ip 
ES )"Gm+1) (2m+2) Bema? 
and 
log T ($7 +1+4 }1i)=log I (Ge#+1) 
=log J 27+ (Be*+4) (log +i) —Be'* 
m Bom 1 1 —(2m t 
SP stare ly (PIT at 
F (r+1) na 
LD ($r+1—$12) C($r+14+422) 
—log / 2x—log V7 + log (cos g)"+1+4 79 tan ¢ 
+(1+7r) log 2 
Bom 1 1 
+S(—1)” (2m +1) (m2) p2m+l 
a — 92m+2 egg 2m+1 cos amelie), 
Let x (7, 6)=4 (the Bernoulli number series in this expansion), 
en: 
Hence : log 
log F (7,1)=log co bie? +log (cos ¢)"+1+7r9 tan ¢+ 2x (7, ¢), 
or, 
EF (7, v)=e-i G(r, v) 
=. / 2 (cos ¢)"+1erbt2x(%d) . ‘ . (iii-) 
i 
Here: 
X(e.= P44) 00) 4 (aye rman. Gr) 
