MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 255 



and that generally, 



We find : 



M (o- 1; r*) = - 



R = 



R = 



<* (R, 



Multiply the first two columns by E r >/R and subtract their sum from the fourth ; 

 multiply by the second two columns by R 13 /R and subtract their sum from the fifth ; 

 divide out by the factor 1/R ; \ We obtain 



m r< 1 



0, 2R 22 , 0, 



0, 0, 2R 33! 



Rl2, Rl2, 0) ~T| 



0, R 23 , R.VJ, -y; 



Now remembering that 



R = 1 rit iii r'f, + 2?- 23 r 31 r,,, 



substitute the various terms, and we find : 



0, 1 - r? 3 , 



dE*. 



dr a ' dr a ' 



dE,, 

 dr,. 



M (o-!, r 23 ) = 



R r> 



0, 0, 



~T **2S^*13i '*12 + 



_j_ r& r l2 , 0, 



0, r a + 



0, 

 0, 



0, - r, ; 

 - ', 0, 



