70 REPORT—1896. 
Calculation of the G (xr, v)-Integrals—Prelininary Report of the 
Committee, consisting of Rev. Ropert HarLEy (Chairman), Pro- 
fessor A. R. Forsyta (Secretary), Mr. J. W. L. GLaisHEr, 
Professor A. LopGer, and Professor Kart PEARSON. (Drawn up by 
Professor KARL PEARSON.) 
APPENDIX  . : . Tables of x-functions, x,, Xx) X53 INA X, - page 75 
. . wv ‘° - 
Preliminary Report on the Integral G (r, »)=| sin’ v9 dA. 
0 
l. Tue integral G (7,7) occurs in the determination of frequency 
curves and of the probable errors of their constants under the form 
e—4”G (r, v), or, what is the same thing, the integral 
je cos” 0 e—v8 dé 
occurs. The calculation of this integral for the values of r, which most 
frequently arise in practice, is for special cases somewhat laborious, and 
this much impedes the use of the generalised frequency curves by statisti- 
cians and biologists.! It seems desirable, accordingly, to form tables of the 
values of the integral for the most usual values of sand». If tan d=r/7, 
then r=2 to r=50, and d=0° to ¢=90° are the ranges of values which 
experience has shown to be most useful for statistical purposes. For the 
same purposes it is not necessary to calculate to a greater degree of 
exactitude than 1 in 1,000. Hence, if a table of double entry be formed 
proceeding by units from r=1 to r=50, and by degrees from f=0° to 
p=90°, intermediate values of 7 and ¢ will be given with sufficient 
accuracy by interpolation ; such a table will contain 4,500 entries, and 
involves a large amount of labour in its calculation. 
The integral G(r, 7) is, however, of considerable interest from the 
standpoint of pure mathematics,” and is not unlikely to be required for a 
variety of investigations, as it is closely related to the Eulerian integrals. 
Hence the formule of this report and the scheme of the proposed tables 
are adapted to expansion, should it be found ultimately of service to form 
as complete a table for G (7, 1) as exists for T’ (x). 
2. The value of the integral may be expressed in terms of Eulerian 
integrals with complex arguments (see Forsyth, Quarterly Journal of 
Mathematics, 1895). Thus: 
2-"r et’T (r+1) ‘ 
fel ot a9 eer'h GS): S 
(0ST ica Grel443) Aad Re 
Bie UT. é amv a 1 ; (ii ) 
r+] Bbrt+l—hui,gr¢ lg hry © 7 
Since e 47” is the mid-value of sind e”®, it is very roughly proportional 
to the value of G(r,v), and accordingly e—#” G(r, )=F (7,1) will be 
found to change more uniformly and gradually than G (7,1), and as this 
1 See a paper on ‘Skew Variation in Homogeneous Material,’ Phil. Trans., 
vol. 186 A, pp. 377-380. A further memoir on the probable errors of frequency 
constants also largely involves the values of G (7, v). 
2 Professor Klein, I am told, has drawn the attention of his students to G (7, vy 
in unpublished lectures, and has suggested to them its fuller consideration. 
