MATHEMATICAL" CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 273 



In like manner we shall now determine the probable errors of the moments and 

 their error correlations. 



Take the logarithmic differentials of (Ixii.) 



p + I 7 





u,; A/J 3A7 



~ 



Pa ~ P + 1 7 



/i, _ 2 (p + 2) Aji _ 4A_7 . 

 ~ y) 7 



Squaring eacli of these in succession and using the known values of -. - y , H ; , y 

 we find 



4 I 4- 



-" " ^ 7(2.7) 1 h i 



Now multiply A//, 2 //x.. by A/x, 3 //A 3 and we find, after some reductions, 



1 + j> + 3 



iy (Ixviii.). 



2 



18S 



Next multiply A^,//*., by A/^//^, and we ultimately have 



1 + 



_ 



4 " "7T77 i _Vi 



V \( f 2S^TiyV\ 



Lastly, multiplying A^ 3 //A 3 by A/i4//A4, w ^ deduce, after some reductions, 



u . bEKjM */ ___ Q xx ) 



-tv.,. u . = ~ ~7^r^7 '/ , ".J\2 \~T /r/ /'9n 4- 3V \1 " " " 



VOL. CXCI. A. 



