(69) 



u's oil the positive side. The iiidicatrix is in the sense u^ ?/,... W/.-fi and 

 if we integrate A'123.../; successively over tiic components of tlic 

 elements of I)oundaiT according to the curved 6//s we find: 



S/-' 



- — dua 

 Ouu. 



JU:, 



ÖX, 



where («/,-(- i «, • • • «/.' = (1 2 3 . . .yj !'^;-|-lj j ; so that we can write 

 as well 



rf 



da 

 du 



-du. 



— dn^ 

 Oh, 



dx 



d>i,/— 



— du,j—i 



dx 



b'((j-\ 



duy—i 



Xn 



]-2^ 



^=l,2...(/- + l; 



0.7;, 



du 



du 



7+1 



dx^ 



7+1 



d»^,+l 



or 



ƒ■> 



123... /y 



1 ^r— du 



du. 



dx^ 



dxp 



dui.^^ 



d.r: 



J' 7 



f/''y,+l 



du. 



'' du. 



da 



du 



— du^jj^i 



dx, 



■p+\ 



du 



du 



7^+1 



y.+l 



If we now move to other parts of the boundary we shall conti- 

 nually see, where we pass a limit of projection with respect to one 

 of the coordinates u, the })rqjection of the indicatrix on the relative 

 curved Cp change in sense. 



So in an arl)itrary point of the boundary the integral is found in the 

 same way as pn the entirely positive side; we shall tlnd oidy, that 

 for each coordinate u^j for which we are on the Jiegative side, the 

 corresponding term under the sign ^ will have to be taken nega- 

 tively, by which we shall have shown the equality of the //-fold 



