40 



AI.MAK NÆS.S. 



M.-N. Kl. 



(14) 2^^,,,,./ d,- 



U-l) 



./.../». ■ ■ • ■■■/■,» 



Jii -i ./";i ./" " 



./ü 1 /2 « J-> " 



y«l/88 7;!" 



./" 1 /" K /" " 



= ^%i^ 12 ^'/■Z' q- <^- cl- 



As obviously the method in Ulis proof is entirely general, this result 

 holds good when we instc'id of 1, 2 have any other two possible numbers, 

 and our theorem is heret)\- pro\ed. 



If 11 = 3, the matrix of the second minors is simply the primar\' 

 matrix. t Let the lattei" be that of the transformation (dyadic) W; then we 

 know that the symmetric- differences in this case are the components of 

 the vector W,.. Vov the thi-ee-dimensional case our theorem thus takes the 

 particular form 

 (15) '/'■ '1', H\- 'I',, 



which simply means that each vector of die triple: 



a — ;^i , b — >;._, , C — y..^ 



is perpendicular to '7'';, [x, conjugate to a, b, 0, a proposition previously 

 stated, ft 



A still more particular case of the same theorem is that if t? is an}' 

 vector in .S'.^, with the three components P, (J, R, then each of the three 

 vectors 



^-V/', p-V,o, ^-Vä 



CX ay dz 



is perpendiculai- to the curl of ü.ttf 



§ 13. Application to Cramer's Rule. 



Let a system of }i equations of the first degree be given, the un- 

 knowns being .v^, .v.^, .... .v,,. 



./1 1 -^'l + .A o X, + . . . . + /1 ;/ .x„ ~ i\ 



... J.^.\\ +/,,, .v., + . . . . +/.nX„ -= V.2 



/"1 -^1 +/"2 '^'2 + 



~r /11 II Xn — Vu 



T though, according to (Al, with tlie elements in another order. This is, however, of no 



consequence as the order of t/j / is altered correspondingly. 

 tt Zur Theorie der Triaden \'oii Al.m.\r N.-ess, § 4 (2I and lal, and § 8 lii. 

 ttt loc. cit. § 4 (el, 



