MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 251 



Divide each row by n, the first, second, third, and fourth rows by cr,, cr,, 0%, <r 4 

 respectively, and the first, second, third, and fourth columns by cr,, cr,, cr :! , er 4 

 respectively. Multiply the fifth column by r vi and subtract it from the first and 

 second ; the sixth column by r, : , and subtract it from the first and third ; the seventh 

 column by r, 4 and subtract it from the first and fourth ; the eighth column by r., 3 

 and subtract it from the second and third ; the ninth column by r 24 and subtract it 

 from the second and fourth ; the tenth column by r :J4 and subtract it from the third 

 and fourth. Remembering that 



rx. = -tin T" ^'12^12 ~T ^*uj"ia ~T **i4-"Hi 



R = R 22 + r.,iR 12 + r 23 R 23 -f r 24 R 24 , &c., 

 and that 



R 12 d /R 13 \ d /R,.;\ d 



12 I;i ,, 



' r " I3 U 





r 12 R ls + '/' 13 R 13 + ?' H R U \ d /R a 



dr,, \~ ~^~ 



A , , 



7l3 "- " 



we find, writing cZ, 2 for d/dr K , &c., for brevity, 



2R n /R, 0, 0, 0, R )2 /R, R I3 /R, RH/R, 0, 0, 0, 



0, 2R 22 /R, 0, 0, R 12 /R, 0, 0, R, 3 /R, R 24 /R, o, 



0, 0, 2R 33 /R, 0, 0, R 13 /R, 0, R 23 /R, 0, R 34 /R, 



0, 0, 0, 2R M /R, 0, 0, R 14 /R, 0, R 21 /R, Rs4/R, 



/R.A , /R^\ , /R\ , /R\ , /R\ , /R w \ , /R\ ., /E\ r/ /Ry\ r/ /\ 



ME"/' 12 U ) 12 \"R ) 12 U/' 12 U/' I2 U/' 12 U/' 12 U/' 12 l R /' 12 v /' 



E.A /R^\ /R^A /E u \ /R w \ /R w \ , /R u \ , /R^X , /E 2 A /E^X 



' I3 "' I3 ' 



/R n \ /EaX /RsA /R u \ /E 12 \ /R w \ ; /R u \ , /R\ , /R 



"B ) Kv H U) M U / 14 (^/' -w j w 14 U/' K B 



/R n \ /R^N /R^X /R 44 \ /R,A , /R,A /E,_A /R\ , /R\ , /B,,\ 



/' 23 U} 23 U/' 23 U /' ^U/' 23 U/ 23 l R /' 23 \ R /' \ K /' \ R /' 



R n \ /KaX , /R^X , /R M \ , /B u \ , /R 13 \ , /B.A , /Bg\ ," /B\ , /B\ 



' 24 ' " ; " ' 24 ' ^ 4 ' M ' M ' E ' 



B w \ , /E.A 7 /B\ , /B^X , /B.A 



AW^^ 



2 K 2 



