IN NON-EUCLIDEAN SPACE. BY A. MC AULAY. 19 



A^' and each of its three named parts crl^', cnv^, grd^' 

 are local when q is local. The operators A, etc., will 

 therefore be called localizing^ operators. 



The reader is recommended to prove the following. 

 In (1) §8 above, w, o- are constant rotors through the 

 origin and w + Jo- =-. w + Jtr' is the velocity motor of 

 a rigid body ; u/ and a being local rotors expressing the. 

 angular velocity and the linear velocity at the point u. 

 With these meanings 



a>' = ^ crl (T, (t' = i J2 crl a>', 



u)' = \ J2 crl^ ,o\ <t' = \ J2 crP o'. 



Since cnv . crl = we have cnvo-' == cnvw' = as 

 might have been expected in the case of a. 



Some additional formulae referring to differentiation 

 are collected here for reference, proofs being left to the 

 reader. In addition to the forms of (5) we have 



e(e + 2J) = A^ - 2Jcrl ) 



= crhcrl — 2J) + grd . cnv + cnv . grd ,S 



in which last again we may write crl — 2J V in place of 

 crl — 2J ; and also we may write 



crP — 2 J crl = Gcrl = crl 9 



= (A - 2J)crl = crl(A - 2J) ... (15) 



The next formula is especially useful for wave pro- 

 pagation of curl ] p being a quaternion function of the 

 point whose position unitat is u 



(9 + J) iiJii). H-' = (A ~ 2JS)p 



= (9 - 2JK)p (16) 



Though 9(9 + 2 J ) is a scalar operator, it appears 

 from the Jcrl in the middle expression of (14) that it is 

 not a localizing operator It should perhaps be noted 

 that though we pay careful attention to localization and 

 often assume q to be local, all our general formulae are 

 true independently of any such supposition. 



