:^8 



AI.MAK N/i:SS. 



M.-X. Kl. 



Wc liere put: 



151 f, - 2;/;, C, i, - l'/,,Ç, i: 1-/.,,^. 



I I I 



and inserting this in (4) we get: 



v/;,e,-2;/,,e,- 2'.A,er 



16) 



^'5//....-^'';/ 

 (//) 



e, e, . 





Å» /-' 



/>,. 



But, according to an eleiiientaiy theorem of determinants, this simply 

 means that (6) can be expressed as a sum of all the ti determinants of the 

 following type : 



71 



Å'^r,f,,Cr 



e.. 



/,, e,- 



/1 :i /m 



./1 " .A " ./"" 



/1 ' /2 ' /" ' 



ei e^ e„ 



/läf-iü /"8 



./ 1 " /2 " /" " 



where especially the subscript / in this case does not indicate a summation 

 in ordinary scnce; it only means that / can be any one of the numbers 

 1, 2, 3 . . . //. And the , .dotted" vector e,- is, of course, to be applied to the 

 „nearest" vectors, i. e. to those in the second row. 



But we now readily see, that by putting / > 3 we get determinants 

 in which two rows of scalars are equal, i. e. : vanishing determinants. Thus 

 we have : 



18) >'gy/,,,./)/-e, 



./m./.x f'-y 



Cj c, e» 



./1 :i ./•-> 3 ./ " 3 



Å"Å" ■ ■ • 



/« 



+ C, • 



./is. /.3 • 



/"3 



/l " J-1 " /" " 



Each of these two determinants is a \ector, whose components are the 

 cotactors of the elements in the second row. If we now expand in terms 

 of these elements (i. e. : in terms of the unit vectors) and then multiply 

 distributively by e^ • and c, • respectively, all the scalar products vanish 

 except one in each determinant, as e, • ey=~-0 1/4^ /) and =1 (/=y). Therefore: 



(9: :^%,,,,<i,, 



./.k/31- ■ 

 f-l 3 ./3 s • 



./«3 



./•J " ./« " /" " 



./13. /3 3 



fr., 

 ./"3 



f.,„ 



