248 PROFESSOR K. PEARSON AND MR. L. N. G. FILON 



n d*(logR) 



~ 



/R\ rf /E 12 N 



" E + * " "' 



Now, since R = R n + S s (Ri s r,,), 



mR = S 



and therefore 



S a ,(R., 



Substituting, we find 



<i ,, i '/i; '/]; 



But (log R) = : - -- and 2R 12 == ^ ; hence 



Thus finally 



where R 12 , r j is the second minor, found by striking out from R the first and second 

 rows and columns. 



In the next place let us differentiate (xxiii.) with regard to r, 3 and sum. We find 

 in precisely similar manner 



fff .. <P lp g s 



j . . . ,<, ^ 



.,._. </i\ 



or 



/i)ff T> T> T> \ 



7 (Jxv 10 ivi 3 ltl\ 12 i.) / \ 



1 .,b 13 = n- - (xxix.). 



Similarly, we deduce 



2E 12 E 3t - 



7, _ ' _ _ i 



2 t> 34 _ n I 



^T \ E I ~ R2 



This completes all the possible types. 



