52 



AI.MAR NÆSS. 



M.-N. Kl. 



the ciiil of a l'ail also he \vi"ilt<n 

 (13) 





tlic sum tf) l)c taken for all possible sets of i, j, when i <^J. 



And exactly in the same way as in § 15 we here prove that the space 

 complement of the curl is equal to (;/ — 2)! times the dyadic whose matrix is: 



c Oj c a i 

 2 .V,- 9 Xj 



which dyadic sometimes is called the curl ot" a. Thus we get: 



<" 2 V Xa = {n - 2)! (V a - (V ^^^-^ -^ ^ti - 21! (V <\-^ V). 



The formula for the divergence of the vector product in ^3 is a particular 

 case of the following equation : 



(15) div a^ Xa,. =- — (— n" 



We have- 



(16) div ai X a,. 



d 



(■'1 • 



2 

 On 





C/ 



<•?.. u 



curl Clj • curl a.j 



a, a.> 



ei 



--(-ij 



e« 



C 



«11 

 ^■21 



C/, 



But this last determinant is obviously equal to the sum of two determinants 



a 



obtained b\- applving the operators - — to the rows a^i and o.^i respectively. 



c .V,- 



The hrst of these clearly is : 



(17) 







2 .v„ 



?! 



Ci • iio . . . . e„ • a... 



9 c 



c'.V„ 



^ curl a^ ■ Cl._, 



