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



It must, however, be remembered that although viscosity 

 does not, to any appreciable extent, affect the forms of these 

 ripples, it does very rapidly damp them out. Thus the 

 amplitude of the ripple in tig. 1 is halved in less than one fifth 

 of a second, so that it must be sought within a very few 

 centimetres of the generating source. But, if the ripples of 

 fig. 4 could be produced, they might be expected to travel 

 some twenty or thirty centimetres without any serious dimi- 

 nution of amplitude. 



LXXIII. On the Operator V in Combination icith 

 Homogeneous Functions. By Frank L. Hitchcock, Ph.D.* 



1. A MONGr the uses of the Hamiltonian operator V there 



xJL are three which are particularly remarkable. First 

 is the use of V to distinguish the character of fields of 

 force, fluid motion, and other vector fields. Second is its 

 use to express integral relations having to do with space- 

 integration over surfaces and volumes. Third, when V is 

 combined with functions which are homogeneous in the 

 point-vector p, many new results are obtained. 



To recall the leading facts under the first category: — If a 

 vector function F of the point-vector p satisfies the relation 

 WF = 0, its rotation vector or " curl " is zero, and its distri- 

 bution is lamellar. If S\7F = 0, the "divergence " is zero, 

 and the distribution solenoidal. If both these relations hold, 

 so that VF = 0, the distribution is Laplacean. If F is 

 everywhere at right angles to its own curl, we have 

 SFVF = 0; as I am not aware of any name for such a 

 distribution, I shall venture to call it orthogyral f. The 

 most significant property of an orthogyral vector is that it 

 becomes lamellar when multiplied by a suitably chosen 

 variable scalar %. 



Under the second category fall the quaternionic forms of 

 the theorems of Gauss and of Stokes on multiple integrals, 

 which have been greatly extended b}^ the late Profs. Tait 

 and 0. J. Joly and by Dr. Alex. McAulay. 



My present object is to develop somewhat further the uses 



* Communicated by the Author. 



t Pronounced ortho jl'ral. 



X Such a characterization of vector fields by means of differential 

 operators may be greatly extended. Thus the four fields to which names 

 are above given are characterized by the linear operators W, Sv, V, and 

 SFV, special cases of the general linear quaternion function of v ? which 

 in these combinations is, analytically, both vector and differentiator. 

 I have considered the general question in a former paper (" The Double 

 Nature of Nabla," Phil. Mag. Jan. 1909). 



