John Blakeman 335 
Problem II. To find the correlation between the deviations due to landom 
sampling in the values of ax and o-j/. 
We have Na^^ = S [iix^ {x^ — xf], 
.-. maM. = S {Sn,^ {.T, - xf} - 2SxS {n,^ (x, - x)}. 
But S [iix {x^-x)] =0. 
Hence 2iVo-^ So-^ = 8 {S/^ {x^ - xf] . 
Again 2NaMSaM = S {Bn,^ {y,^ - yf] + 2;Sf [hj,n,^ {y,^ - Tj)]. 
Multiply these two expressions together, sum for all random samples, and divide 
by the number of such samples; then using (iii), (vi) and (vii) we find 
Hence 4a-.^(rjj-S.,^S.r^^ig.^.^,^ = "^"^ (ix). 
Problem III. To find the correlation between the deviations due to random 
sampling in the values of p^^ and cm- 
We have = S' {%^,^^. {x, - x) (y, -y)}, 
Mpn = S' {Bn^^y^ {.Xp - x) (y, - y)} - BxS' {y, - y)} - SyS' {^i.^^y^ (x^ - x)}. 
Hence NBp,, = S' [Bn^^y^ (x^ - .?) (?/, - y)]. 
But 2Na3iBaM = 8 {Bn,^ (y,^ - y)"-} + 28 {By^n,^ (y,^ - y)]. 
Multiply these two expressions together, sum for all random samples and divide 
by the number of such samples ; then using (iii), (iv) and (v) we find 
2i\^V3f 2p„2<,j^i?p„<.j^ = - -^Pj^- + S' [n^^y^ - x) {y^ - y) (y^^ - y)'} 
+ 2'S' {n^^y^ (x, - x) (ys - y) {y, - y^^) (y^^ - y)}. 
But 
S' {'Vs («P - iVs - y) {yxp - yf] = (n^^,,^ {x, - x) (y, - y^) {yocp - yf] 
while S{nxpVs(ys-yx)] = ^- 
Summing with respect to s and keeping p constant. Hence 
^' ['>\ys (^p - ^) - y) (y^, - yf] = . 
Again 
S'' [nx^y, («P - x) {ys - y) (ys - yx) {yx^ - y)\ - ^S" {x, - x) (y, - y^)"- {y^^ - y)] 
+ ^' {'%ys (^p - - y^) iyx^ - y)% 
Sum with respect to s, keeping p constant. Since 
'S'l'V..(ys-^x,)j =0, 
and 8 [n^^y^ (ys - yxlT] = n^^ a,,^ ' , 
