26 



AI.MAR NÆSS. 



M.-N. Kl. 



another ni(|( i, .uhI, ihcrcforc, the nmltipHrati«')!! of.scalars being commutative, 

 the prodiici ol' ih(,- scalars is e(|ii,il in eacli of them. Moreover, each term 

 having- the sij^ii oj" its pciniiilalif)!), il has the sign + or — according as 

 it is an e\(n or odd [xiiniil.ition of the first one (those terms cancel out 

 where two ol the a's are (iiikiIi. I leu« < the sum of all these terms is 



(K) 



Ca, Ca., Ca.p 



e«, Cr, 



.Up 



(l^ (ti CI., (I'i 



Ip Up. 



And tile expanded determinant (13) is then reduced to the sum of all 

 terms of this form, tlie a's ixing any p numbers in a/iy order of 1 , 2 . . . . ;/. 

 But then there will be /> ! of these terms which contain the same set of unit 

 vectors, but with the coliiiiiiis of the determinant in different order. In one of 

 them tlie indices will occui" in order of magnitude, say k^ <^ k.^ <l . . . . <^ kp, 



the term accordingly 



(U 





. . e/,. 



^k. 



"i Ai «^o ki • ■ ■ ■ ^p k 



And as the columns of a vector determinant can be interchanged as 

 in an ordinary one, all the others can be written : 



(M) 



+ 



Cfei et2 e^ 



Ca-i eA-2 . . 



Ca- 



1 /5l ■ip<> 



where the /i's stand for all permutations of the k's, the sign being deter- 

 mined as usual. Hence the sum of these p ! terms is : 



(N) 



And, consequently, the vector determinant in (13) is equal to the sum of 

 the I ) terms of this form, from which follows the desired result. Formula 



(5) is thereby proved. 



If we in (5) put p = 2 we get the more special formula (4), which 

 also can be written : 



(14) 



a X« - 2 (b X c) -- — (;/ — 2)! a • [be - (b o 



