516 



Here we mean by c that index, wliicli with a and h forms an 

 even permutation of I 2 3. By pqr we shall denote a similar set: 



abc{=)pqr{=) 123. 



We already saw that displacements of this kind constitute a rota- 

 tion. To inquire whether the angular amount of the rotation is 

 equal to the length of 1, we must observe the displacement of the 

 end of a vector u which is perpendicular to 1, so that 



2 {ah)gabu<'l''=^ (9) 



This displacement ought to be 'u multiplied by 1 . Let us cal- 

 culate \'QY ■■ 



I 9ai gu gd 



-— {rij V to) I gaj gbj gcj 



gnrgOrgrr 



li VJ 



öl'V g qo 

 g/nr gqir 



I'U "'. 



The summation with respect to r has been etïected by writing 

 in full the tirst determinant, if we want to sum up with respect 

 to r, we notice that the determinant vanishes bul for one special 

 value of ?', wiiich is different both from / and ƒ If i = p, j = q, 

 then the determinant becomes -j- y, if / := y, / :^ />, then we get — g. 

 In both cases we may write 



C,^= ^{pqvxo) 



Pr-gr 



I J' /'• ?<y n"\ 



g/nrg(jw 



and, by (9): 



' - $' = n* l\ 



So the correctness of formula (8) has been shown, 

 But then we are justified in putting 



\/g 



gai ga 

 gnj gij 



l\ 



and we can subsequently show that (7) represents four times the 

 parallelepiped mentioned. We Avrite (7) with a slight alteration of 

 indices and we get : 



2 :S (c ./■ k) gclc Rj {dxj 6xK —dx^öx J) = 



gai gbi Oci 



Vg 



^Oi^) i g»j gbj gcj i l^ {<ivj (fxJc — dxköxi). 



gak gbk gek \ 



Now if j and /; assume all values, a set j, k furnishes just as 

 much as a set k,j, the determinant taking the value -\-g or — g 

 according to the combination i,j,k being an even or odd permutation 



