MATHEMATICAL CONTRIBUTIONS TO- THE THEORY OP EVOLUTION. 261 



Now b*sA t p^- , M&e^ , . . . <!!*'<> , are the correlations between the errors 

 AS^ AS^S,, A2JS, A2., U 2, J4 



made in the various quantities o-j, cr 2 , o- 3 , c7- 4 , r J2 , r, 3 , r u , r 23 , r 24 , r 3 ,, every one of which 

 is known by the previous investigations except that of r, 2 and r M or M ( ,. ui ...^/AS,.^,^. 

 Hence the above equation will suffice to find the latter quantity, since 2,. u and S,. M are 

 known. We have 



1 



~.=if(' + !).j 



n RI^'H 7/. Ii 2 ,n, 



T> > "'<T 4 <7. ~~ T> J 



rj(7j J.X CTjiT.) It I 



by (xxiv.) and (xxv.), 





Further, 



M ( , v , .,) _ 

 -~ 



^i^> = ji r,, (1 - rr,, by (xvi.) and (xviii.), 



by (xxxvi.), 



- ^ {* (' - r 12 r S3 ) + r. (r a - r B r u )} 



a j- 



+ r M ( - r^u, 



J 



r; 4 ) - ir^K^}, 



). by (xxxvii.), 



/^,) - *V1U, J 

 - r' 34 ), by (xvi.). 



J Iv 



Now substitute and divide out by common factors, and we find 



(v^Kii + 'VJU r lz (l - ??,) + r u ll u {r w (r a - r,,r,,) + r a (i: a - /',. 

 + (R + E 44 ) {r, 4 (r, 4 - r M r, 2 ) + r u ('/.> 4 - r,,r u )} 

 + 2K 14 {' M (1 - r? 2 )(l - 7-i 4 ) - J r I2 r u R 33 } 

 4- 2K 24 {(! - r? 2 ) (1 ri) -^r^r^R^} 





