The Double Nature of Nabla. 113 



a whole, while the velocity V o£ compressional waves is 

 given by 



*=»*km (102) 



These expressions are given merely by way o£ illustration; 

 they are o£ course far from being the most general, in the 

 first place because of our assumption that all the vortices 

 may be treated as linear, and in the second place because it 

 has been supposed that the core of any one vortex either con- 

 sists wholly of rotationally moving liquid or is wholly vacuous. 



V. The Double Nature of Nabla. By Frank Lauren 

 Hitchcock, Kenyon College, Gambler, Ohio, U.S.A.* 



fT^HE importance of the operator nabla in Physics must in 

 JL the nature of things continue to increase. The more 

 we come to regard a force between two material bodies as a 

 manifestation of the properties of the intervening medium, 

 the more any attempt to express facts in mathematical form 

 leads to the use of the potential, and thence to nabla or some 

 equivalent of nabla. Every actual distribution of force has 

 a potential, which is sometimes a scalar, sometimes a vector. 

 in general a quaternion, but in any case yields the force again 

 when acted on by nabla. The same is true of every distri- 

 bution of velocity in a fluid. Nabla might even be defined, 

 from the point of view of a physicist, as that operation by 

 which any quantity is derived from its potential j". 



It is therefore of interest to inquire what are the most 

 convenient methods of working with this operator and applying 

 it to various kinds of functions. As is well known, nabla is 

 a differentiator, which we may, if we wish, express in terms 

 of three ordinary differentiations ; and is also a vector, in a 



mathematical sense, obeying all the laws of: vector algebra. 



. . . 



We arrive at very direct methods of handling nabla as soon 



as we distinguish clearly these two properties, and take full 

 advantage of the transformations to which they lead. 



For, in the first place, nabla, in its capacity of differentiator, 

 conforms to all the familiar rules for differentiation so long- 

 as we do not move it from its place relative to other vectors. 

 Vector multiplication is not commutative, but it is distributive, 

 and, in its original form, it is associative, obeying, therefore, 



* Communicated by the Author. 



t For a proof by the late Prof. C. J. Joly that every distribution of a 

 quantity has a definite potential, see his appendix to Sir W. K. Hamilton's! 

 ' Elements of Quaternions,' 2nd ed., p. 451. 



Phil. Mag. S. 6. Vol. 17. No. 97. Jan. 1909. I 



