﻿704 Dr. F. L. Hitchcock on the Operator V in 



are not analogous facts in regard to lamellar vectors. The 

 following is the case : — 



Any homogeneous vector may be rendered lamellar by 

 adding to it a term of the form Y pr, where t is a properly 

 chosen vector ; exception must be made of vectors of 

 degree —1. 



For consider the well-known vector identity (Tait, Art. 90), 



V*Vi8y=yS*0— £Say. 



Writing p for a, V for j3, and Fp for y, this identity 

 becomes 



YpYVFp = F P ' .8pV-VSpFp\ . . (10) 

 where, on the right, accents indicate that V acts only on Fp. 

 By Euler's theorem, Fp' . S/>V'= — nFp. Also, 



VS^F/^VS/^ + V'S^V, . . . (11) 



by distributing V- (Unaccented V acts on all that follows 

 in the same term.) But the first term on the right of (11) 

 is the same as —Fp, by Tait, Art. 146. Whence (11) 

 becomes 



\7SpFp=-Fp + V'SpFp ! (12) 



By adding (10) and (12) and solving for Fp we therefore 

 have the identity 



YpYVFp _V$pFp 



9 n+i n+1 ' • • • y 16 > 



Fp being any vector homogeneous in p of degree other 

 than — 1. The right-hand term is obviously lamellar. 

 Stated in words, (13) shows that any homogeneous vector 

 field (exception noted) may be taken as the sum of two 

 fields, one lamellar (irrotational), the other at right angles 

 to the point vector. By transposing, and operating with V V, 



(13) becomes 



•vv{*, + ^}-o. . . . (U, 



This latter identity proves the proposition, and shows how to 

 find the vector t. The identity (14) may be verified by 

 direct operation*. 



* The method used above for obtaining (14) is not quite parallel to 

 that by which the analogous (9) was proved. Indeed, (9) might have 

 been proved by first establishing the identity, analogous to (13),° 



(n+2)Fp = -pSVFp-YWpFp, 



by expanding the last term on the right by the formula, Phil. Mao-. June 

 1902, p. 579, (6). We then have (9) by the operator SV ; or we have 



(14) by writing VvFp in place of Fp which is any homogeneous vector 

 so making n become n —1. 



