48 



ALM AR NÆSS. 



M.-X. Kl. 



The C(|ualioii i:j S (SI can also be r)|)taiiu;(J frf>m this by the following 

 theorem : 



(a) ihr space (■niiiplcmoit of any nuinhrr of vectors (say p) is cqual to 

 the scahr product of tlie first vector l>y the space coiuplemait of the others, 

 talceii with Ihr si<^;/ ( 1)" '' . 



Let P, he a pohad ot" ordei- r. Then the theorem says: 



12) 



VX' M=(- 1)" ' 'l-<' P.. 



It is easily proved. Let P, — bj . . . . b, (it is readily seen that the 

 proof is valid also in the case that P, is a sum of such pol3'ads). Then: 



^X'Pr 



13) 



(-1)' 



ei 









''Vi 



/v„ 



= (-ir 



i^ • <' p, 



As ^p a^ . . . Cl;, is of order n — /> , we can put : P, - ^pcI, . . . Clp = Pi; - p, 

 and inserting this in (12), we get immediately from (11): 



(14) \? X" - /^ (<P ^li 



ao)= 



1) ' \n -/))! V- 



^1 



If we in (12) put P, = a, we get 



(151 • i^X« = (- i)"v-<a 



which by § 14 (5) can be written: 



(16) »Xa -= ü-(/Xa). 



The well-known equation of the same form in S.^t is thus valid unaltered 

 in Sn. In So the equation is self-evident, t» X »■'^ t'"'^''' simply means the 



t GiBBS, Scientific Papers II, p. 59. 



