MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 259 

 Expanding this determinant from its first column, remembering that R, 3 R n R 13 R 12 



M 





 ri,, r 13 ) = -- , 8 , T>5 { r. a n (R r 23 R 23 -f R 22 R a3 ) + Via (R,,R n RfzJ 



WJWJWJ-MP 



Or, since R 22 Rn R'^ = R, R 12 R J3 R^R^ = r, 3 R, and R r, 3 R, 3 + 

 = 2R, 2 R 33 i\.,r^ R, 3 , 



M (r la> r ia ) = -7^7, {2A 3 R,,R 33 - r B r u (R M + v, a ll 2a + r^U,,)], 



o jtr.)CTyit 



__ !^L- {,. li T? _ i^j? - 

 - fffa|aill l I " M 2 



Hence, since 



T> _ ^-....V 



Lt "*"> ~ AS, 2 S, 3 ' 

 we have by (xvi.) of Art. (6) and (xxxiv.) of Art. ( ( J) 



TJ - ,, . 1 ,. ,, 



,= 'i 





/V'/'i.., (1 '4; ''Is ''i- + 2/wr,.,) 



- -- - 



To complete the theory for the errors made in an investigation of the constants 

 for a system of three correlated organs, we require to determine the probable error of 

 a regression coefficient for a partial regression of a first organ on a second, the third 

 organ being constant. This coefficient is given by 



Take logarithmic differentials 



Spi g _ ____ 8js_ r a Sr a ^ L^~Jjs __ J^ 



sPw " ~ ''is Vis r a - r * r i3 !3 Vis - '^ r i3 1 - r l 



Let this be squared and divided by n, and then the values found above for the 

 standard deviations of the errors in r n , r n , r., 3 , o-j and <r,, and for the correlations of 

 errors in these quantities be substituted. After some lengthy algebraic reductions, 

 which it seems unnecessary to reproduce, there results 



2 L 2 



