( 68 ) 



Tjet US suppose this determinant to be developed according to tlie 

 first column, let us then integrate partially- each of the terms of the 

 development according to the differential quotient of JTios... .y, , appearing 

 in it; there will remain under the (/>-[- l)-fbld integration sign 

 P ip ~\~ 1) terms neutralizing each other two hv two. Thus for instance: 



du. 



diijj^i 



X 



and dn. 



<^"y;+l 



X 



d\v. 





123.../) 



\-23...p 





dii^dujj^i 



d\v 



dxn 



du. 



bu^ 



b.vp 

 blip 



b\t 



bu^bup^i 



as they transform themselves into one another by interchangement 

 of two rows of the matrix-detei-minant. 



So the y;-fold integral i-emains only, giving under the integration sign 



X 



I 23 ... ;) 



bxj^ 



1 ^-dii, 

 Oil, 



± 1 



b.v 



bup-\.[ 



'-dupj^i . 



b.vp 



— dll^ 



bxp 

 biip^ 



- ^'//)-l-i 



to be integi-ated over the boiindai-y, whilst in a definite point of that 

 boundary the A"^^ term of the first column gets the sign -f- when 

 for the coordinate ?ip the point lies on the positive side of the boundary. 

 Let us now liud the integral of X123...;) over the boundary and 

 let us for the moment suppose ourselves on the part of it lying for all 



