214- Dr. L. Isserlis on the Variation of 



(ii.) Let u, v, w be the mean values of r 23 , r sl , r 12 in many 

 samples o£ size n out of an infinite population with normal 

 distribution. Let X be the mean value of R123 as calculated 

 in such samples, and let w, v, w, X be the corresponding 

 values for the sampled population itself. In any particular 

 sample the correlation coefficients will have values that we 

 may denote by u + du, v + dv, w + div, X + dX. 



Now suppose that w=w+a, v = v + /3, w = w + y, X = X + f, 



then a, /3, 7 are known correct to terms in -s, viz. : 



n z 



_ u(l-u*) 3u(l-u 2 )(l + 3^) 



2n ' n 2 -' • UO 



N ° W v / fv 2 + w 2 -2uvw\ ,, 



x= V ( — i^? — / = ^' v ' w) sa7 > • ( 18 ) 



and X + dX =f(u + <fa, « +" «fo, w + dw), 



X-ff+dX=/(w + a + dw, a+^ + dto, ^ + 7 + ^). (19) 



Expanding the right-hand side by Taylor's theorem we 

 have, after subtracting (18), 



S + dZ = (* + du)M+Q3 + dv)l£ + (y + dw)& 



+ 2(« + du)(^ + ^)^ +2(^ + ^X7 + ^)^ 



+ 2( 7 + ^X" + ^)J^l 

 OWQUJ 



+ terms of higher order in du, dv, die . . . (20) 



Let us sum for all samples and divide by the number of 

 samples. We shall have 



OU OV c)w 



+ 2(«0 + cwj § g- c + 2(/3 7 + <ww) ^ 



+ 2(7a + o-«,a"„r wlt )^-^- >+ etc (21) 



* H. E. Soper, Biometriha, vol. ix. p. 105. 



