MATHEMATICAL CONTRIBUTIONS TO THE THEORY" OF EVOLUTION. 253 

 In this case R = 1 r|, rj, r?., + 2r. a r 3l r K , and we have 



1 - r? 3 , R 33 = 1 r\,, 



whence we have 



A _(rJ^A; 2 (!-<), 0. 0, V- ; , r l4 r a -- lfc 0, 



ttl R K 0> 2(l-ri,), 0, r w r u -r M , 0, ,- 31 r, 2 - ,- 23 , 



0, 0, -(1 >!>) 0, /Yz'-'s r, 3) r,,-;-,, ),.-,, 



. > -'V' 1, ''a, v-31, 



. 2r ig , 0, r, 3 , 1, r,,, 



-2r., 3) 0, 0, r. fl) r, 2 , - 1. 



Divide the first three columns by 2 ; multiply the last row by r yi and subtract 

 from the first row ; the fifth row by r, 3 and subtract from the second; the fourth row 

 by r a and subtract from the third ; we find 



column, it again reduces by one degree. Repeat the process twice more, and after a slight rearrangement, 

 the fact that d (R> pp )idr fg = being remembered, \vc have, if . = R/V'R M /.R r>l 



A.IO *?*R Ti T? 1-? 



d * d i cl> i /"/ , 



57" 8 '^" gjr lo gm JT ^ *'" ,77~ sf: " " 



<i i '? , d , 



-r- log ft. , j- log ft, , log f,, , 



wa 7 1S d) j-j 



^ log f,., , _ log f, 3 , 

 ar u rt?- ]S 



^lofffi., 4; log ft,, 



rf , 

 ^logf !3> 



rf , 



J^ log ft : . , 



the general run of terms being obvious. 



In precisely similar manner the value of A for p organs can be written down, its degree being p Jess 

 than the form given in (xxxiii.). We have not succeeded in reducing A for the general case [since 

 writing this, Mr. ARTHUR BERRY, of King's College, Cambridge, has succeeded in reducing the deter- 

 minant for p = 4, and also in showing its relation to elliptic space], but we feel fairly confident that its 



value will be found to be - 







0(0} . . . <r f 



