32 



A I. M AK NÆ.SS. 



.M.-.\. Kl 



Tilt- Krgän/.ungs.svstcm has a few properties which may be worth 

 iiotin.i;. t Here we sliall only mention that the Ergänzungssystems of two 

 conjugate vector systems are conjugate. This follows from : vv 



where y denotes the (Gibbsian) double cross product. 



From our point of view, the Ergänzung of a vector of a system oi 

 ;/ vectors in .S« must be the s/)ace coinplcmcnt of all the others, taken 

 alternately with positive or negative sign. We will give the definition the 

 following form : 



(a) The i"^ Ergänzungsvedor of a given vector system f, is obtained by 

 striking out the i"' row in the déterminant of the f's and replacing it by 

 the unit vectors. 



If the /"' Ergänzungsvector is denoted by XO,, we get: 



(2) 



\V, 



Ai /i 



A» 



fi - \ \ fi- \2 /' - 1 " 



Ci Co e„ 



/' + 1 1 // + 1 2 /'■ + 1 " 



/m fm . 



. . ./»„ 



e« F,„ = e/ F,j 



where F,j is the cofactor of /, /. We thus see that the matrix of the Er- 

 gänzungssystem is the matrix of the cofactors, i. e. conjugate to the adjoint 

 of the matrix of the f's. 



Now (2) evidently can be written : 



(3) 



W, 



<"- 1 fi . . . f - 1 f+l . . . f,r 



It is now easily shown that the reciprocal system of the f's is deter- 

 mined by the n equations : 



(4) 



f- = ^«>, 



analogous to what we have found in S.^. \f\ is the determinant of the f's. 

 Let us put : 



(5) - e, f, ; <P* = e, f* ; 0c* = \7 e,. 

 Moreover: 



(6) f* f = (f * e,) • (e/ fy) = 0c* • 0. 



t loc. cit. § 13, § 37, § 46. 

 tt loc. cit, § 12 I5) and § 13 (i). 



