1898,] 



of a certain determinant, etc. 



the work to replace those variables r which have suffixes not 

 exceeding n — 1 by other variables p where 



pij = (Tig - r in r jn )/ V(l - r\ n ) (1 - r* jn ). 

 This substitution will be denoted by the notation [9fc] r _ p &c. 

 If in the determinant 



R = 



1, r n , ..., r- 



Vm > Tin > •••> J- 



we multiply the last column by r^ and subtract it from the ith 

 column (i = l, 2, ... n— 1), we get 



*■ ** m> ^*i2 finXmn ^is Ti n V 3n , • • • ^m 



12 ' m ' mi > x ' 2»i ? • • • • • • 'in 



r \, n—i 1\n Vn—i, n> 



o, 



o, 



1__ /V*2 /V* 



' n— i, »u ' n— l, 



0, 1 



-L *" m> ^12 ^*in*'2?i> ••• *"i, n— 1 f\n^n— l, n. 



'12 » 1« ' BM.I - 1 - ' 9«, , • • • 



I Ti,n— l Tin Tn— i, ni — ••• *■ ' n— 1, n 



Dividing out the ith column and the ith row by 



^l-r\ n (t = l, 2...W-1) 



this becomes 



i=n— 1 



n (l-r 2 iM ) 



1, Pis, 



Pl2, 1, 



Pi, n-i 

 P2, n-i 



So that 



Pi, n— l 



i=u— 1 



i2=II(l-rW).[9t] r= , 



i = l 



(7), 



with the notation just explained. 



A slight modification of this work enables us to express any 

 symmetrical determinant of order n as a symmetrical determinant 

 of order ?i — 1, a result which is probably well known but which I 

 do not remember to have seen before. 



