24 
PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 
and, generally, whether s is or is not = s', or these are or are not = p and q, we 
have 
d /R, 
d v.pq \ R 
Ejm R ijs 1 + Ii/is' R 
■pS’ 
R 2 
(lxxi.). 
This follows thus 
d / I tv' 
d^‘pq \ R 
1 ^/Rj Rjj/ dR 
R dr n “ IP dT 
1 dR, 
Us' 
Vi 
R dr 
n 
2R^< R pq 
Pd ’ 
or we have to show that 
dlR. 
2R m , R pq I * s R^i' R/«' Eji 
dr 
n 
R 
Rys Ryi' 
R 
— pij^ss 1 T" 
R m / E pn 1 1 ; v R y s 
R 
where ^R^ is the minor corresponding to the term r pq in If,,,, and ^R^/ the minor 
corresponding to the term 
w 
But tliis last result is obvious because only con¬ 
tains r pq in two places, i.e., as r pq and r qp . 
Putting s =s', we have the other identity required above, i.e., 
d (Rss 
dr n \ E 
2Ryu- Ry^ 
R 3 ‘ 
1 d; 
Returning now to the value for - on the previous page, we see that the two 
Z CtTpq 
sum terms may be expressed as a product, or we may put 
- — — _ 5k ifi f^-x \ y S & r 
zdr n ~ R + 1 1 R 7 X bl \ R Xt 
Now write 
N 
z = 
we 
(2,r)VR 
—(/> 
Then 
— g f^i s x \ c l^_ — g ( x \ an q _ ^ 
dx p ~ R x *y dxq ~ bl i R 7 and dxplx.q 
_ Ryr 
Plence 
1 dz _ d~<}) d(f) dcf> 
z dr n dxp dx q dx p dx q " 
Now differentiate log 2 with regard to x p . Then 
dz d<p 
dx n dx. 
y 
* See also Scott, 1 Theory of Determinants,’ p. 59. 
