1922. No. 13. ON A SPECIAL POLYADIC OF ORDER // — p. 4 1 



Let US here by r denote an unknown vector, r ^^ C, .iv, and by \? the known 

 vector \? — e, tv. Putting, moreover, f/ =^/1 d, then (1) can be written: 



fi • 1^ = î'i 

 f> • r =- l'a 



(2) 



f« • jr ^ v„ 



Multiplying these equations by e^, e._, . . . . e« respectively, and adding, 

 we get 



(3) Ci t\ • r - c, f, • r + ■ • • • + e» f„ • r = ü 

 or 



(4) . r = t? . 



To solve the equations 1 1 1 then simply means to find that unknown 

 vector r which by the known d3'adic ^ is transformed into the known 

 \ector ï>. We know that the equations (1) are always solvable if the f's 

 are not all contained in a subspace, Sp, of S„. For in this case ø • r will 

 also be lying in a /»-space, viz. the /»-space which contains the conjugate 

 vectors to the f's, and which in general is différent from Sp. 



Now (II is solved by multiplving (4) bv 0,*, 0* being the dyadic 

 determined by the trciprocal sysleni of the f's. From (41 then we get: 



(5) 4>c* ■ (Pï ^ 'Pc* ■ X! 

 which reduces to 



(6) ):=0.*t?. 



This single equation involves Cramer's formulae. Let IV,'^ be the Er- 

 gänzungssystem of the /<'s, i. e. the conjugate system to the Ergänzungen 

 of the f's (see § 101. Then: 



(7) 0/ = e,îî^ 



I/I 



and (6) may be written: 



(8) ,,,,= ,,^ = ,,^ 



The components here being equal each to each, we get: 



which are Cramer's fornmlæ. We notice that the space complement in this 

 very compact formula serve to determine the unknowns exactly in the 



