RELATION OF VARIABLES. 109 



Ri = {S(.-l^ - A) U- - Xi) + dZix - Xi)4 +/2(.r - xOe 

 - R^iy - 74)1 (.1- - Xi)4 - RiZiy - y5)i(.r - Xi)o 



- ReL(y - yM-r - xOe} 



2(x - Xi)2 

 Similarly equations (24) to (28) are derived 



(23) 



R2 = (2(5^ - B) {x - X2) + dZix - X2)4 +/2(.r - Xo)6 - Ri^^iy - 74)2 



{X - X2)4 - i?52(.V - y5)2 (.r - X2)5 - i?6S(?/ - y6)2 (.r - X2)6} 



S(.r - X2)2 



(24) 



i?3 = {2(C^ - C) (.T - X3) + (C(.r - X3)4 +/(.r - Xgle - i?42(7/ - 74)3 



(x - X3)4 - Ro^(y - 75)3 (.r - X3)5 - R6^{y - ye)s (x - xs)<i} 



S(.T - XsY 



(25) 



i?4 = {2(I>^ - D) (2/ - 74) + aS(^ - y4)i + cS(^ - y4)3 - i?i2(.r - Xi)4 

 (y - y4)i - i^22(.T - X2)4 (.?/ - y4)2 - Rs^jx - X3)4 (t/ - y4)3} .26) 



R, = {S(£^ - E) (i/ - ys) + a^iy - y,)i + c2(^ - y5)3 - Ri^x - x,), 



(y - y 5)1 - R22{X - X2)5 (.V - y5)2 - R^JX - X3)5(y - y5)3} ^r,J^ 



s(i/ - y5)^ 



Re = {S(F^- - F) (7/ - ye) + a2(7/ - ye): + cZ{y - y,), - i?,S(.T - Xi)6 

 {y - y6)i - i^S(.r - X2)6 (,;/ - ye)- - Rs^ix - xs)^ (y - ye)^] .^c^ 



For brevity, let 



Mii = S(.r - Xi)4, J/16 = 2(x- - xOe, etc. 



P41 = 2(^/ - y4)i, P43 = S(t/ - y4)3, etc. 



Xi4 = S(.r - Xi)4 (.?/ - y4)i, Ki5 = 2(0; - Xi)5 (y - y5)i, etc. 



Li = 2(0; - Xi)2, i^ = 2(.r - X2)2, etc. 



Finally, in group 1, for example, the observed average of w, A\ may be 

 substituted for the required average A, in the expression 2(^1* — A) 

 {x — Xi) because the sum of the deviations {x — Xi) is zero. In other 



