1922. No. 13. ON A SPECIAL POLYADIC OF ORDER // — />. 



51 



According to what is said above, we get: 



181 



V-iV Xai = 



e, e., . 



c x„ 



K\ e. 



c X, c .v., 



On 



which vanishes identically. Hence the curl of a, defined as we have done 

 above, satisfies the characteristic equation 



191 div curl a ^0. 



We also find : 



(a) T/ie (divergence of the space coiuplenieiit of a vector is equal to the 

 curl of the same vector times (— 1) . 



For rememberine that the e's are constant \ectors, we get 



101 ■!<v1) 



9 



a .V;, 



(— 1) 



ei 



e, 



a. 



'1 



C;, 



a„ 



from which the proposition follows. This may be written : 



111) V •<a = (— n" V X a 



and in this form it can be regarded as a particular case of § 15 115), 

 "0 being interchanged with Xabla, and r — 1, i. e.: P, = CI. 



Also in (9) (or (811 Nabla plays the rôle of an ordinary vector, as 

 V • Iv X ai vanishes identically too. 



Bv § 14 151, (11) can be written: 



(12) 



V -f/X ai = V Xa 



which is only a special case of § 15 (161. This equation is well-known in -So.v 

 By § 9 (141 (151 and remembering that in this special case: 



/• • = ^' 



f Zur Theorie der Triaden von Al.mar X.ïss, p. 121. 



