510 Disti'ibution of the Correlation Coefidents of Samples 
The point in 2n dimensional space which is represented by the n pairs of 
observations must be such that its projection on the n dimensional space, in 
which X is represented, lies upon a certain sphere of radius fi^ sjn, and on the space 
in which y is represented, upon another sphere of radius iJL^s/n, and now, when we 
come to the interpretation of r, we must observe that to each point on the first 
sphere there corresponds a certain point on the second sphere, to which it bears 
the relation 
yi-y~ y^-y~ "'~ yn-y' 
In general this relation does not hold for the n pairs of observations, and the 
two projections will not fall at corresponding points on the two spheres. If now 
one of the spheres be turned round so as to occupy the same space as the other, 
and so that the lines upon which and t/i, and the other pairs of coordinates, are 
measured, coincide, then corresponding points will lie on the same radii, and the 
correlation coefficient r measures the cosine of the angle between the radii to the 
two points specified by the observations. 
Taking one of the projections as fixed at any point on the sphere of radius jx^, 
the region for which r lies in the range dr, is a zone, on the other sphere in ?i — 1 
dimensions, of radius /ZiV?iVl — r^, and of width /^j c^r/ Vl — r^, and therefore 
having a volume proportional to — r^) dr. . 
4. We may now turn to the direct simplification of the expression (I), at each 
stage discarding any factors which do not involve r. 
g 2^1^ 2<ri<r2 2<,2' dx,dy,dx^dy, . . . dx^dy^ 
may be reduced to j 
^ l-p2-) 2<ri2 2(ri(72 2^^^ \ 
dxdyiJi{^-^dix^fi^^-^dfjL^{l-r^) ^ dr, 
n_ { 1^ _ 2p'-/^i|U2 _^ ) w-4 
or to e l-/''*2<ri' 2cri<r2 (i _ 2 
In order to integrate this expression from 0 to oo , with respect to fi^ and [jl^, let 
and we have 
/"oo ro 
dz\ 
J -00 Jo 
^'-'d!;.e .(1-?^=) ^ dr, 
/o (cosh.-pr)"-^ -^^-^^> ' 
