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



7. Most of the foregoing results are extensions of familiar 

 properties of linear vector functions. The identity (13), 

 expressing a homogeneous vector at the sum of an irrota- 

 tional vector and a vector perpendicular to p, appears in the 

 linear case as the familiar 



<J>p=a>p+Y€p (Tait, Art. 186), . . . (17) 



where co is a self-conjugate linear function and e is a constant 



vector. Here cop= ^ P 9 , and Vep = ™ % , whence 



the identity (17) may be written 



<Pp= i£^p + i£+^- .... (l8) 



To bring out the analogy between (13) and (18), put, as 

 before, dFp=<j>dp, so that cf)p = nFp by Euler's theorem, and 

 VSpFp=-cf>'p-Ypbj (12). By (2),V/)VVFp(f-fl/). 

 By substitution of these values, (13) becomes 



* p ~ n + 1 ?z^+T) * ' * W 



which evidently reduces to (18) for the case n=l. The 

 right-hand term may be taken as an extended cop; and, just 

 as cop is at all points of space normal to the general family 

 of central quadric surfaces ^>pcj)p = const., so this term is 

 normal to the cubic or higher surfaces S/oF/3 = const. Again, 

 just as the axes of cop possess the special property of being 

 mutually at right angles, so the axes of its analogue have a 

 specific configuration ; but the consideration of axes lies 

 outside the scope of the present paper. 



8. The identity (13) also throws a good deal of light on 

 the nature of orthogyral vectors. To distinguish these 

 sharply from other vectors, we may say that an orthogyral 

 vector is one satisfying the two following conditions : — 



1. Neither the vector nor its curl vanishes identically. 



2. The scalar product of the vector and its curl vanishes 

 identically. 



Lamellar vectors are thus excluded from the company 

 of orthogyral vectors. It will also be convenient to distin- 

 guish two cases, according as SpFp does, or does not, vanish 

 identically. If an orthogyral vector ¥p is everywhere at right 

 angles to p, the family of surfaces normal to Fp consists of 

 cones. The right-hand term of (13) disappears, and the vector 

 may be said to be conical. If, on the other hand, SpFp does not 



