OF PHYSICAL QUANTITIES. 263 



We may remark that even here, where we may seem to have reached a 

 purer air, uncontarninated by physical applications, one vector is essentially a 

 line, while the other is defined as the normal to a plane, as in all the other 

 pairs of vectors already mentioned*. 



Another distinction among physical vectors is founded on a different prin- 

 ciple, and divides them into those which are defined with reference to translation 

 and those which are defined with reference to rotation. The remarkable analogies 

 between these two classes of vectors is well pointed out by Poins6t in his 

 treatise on the motion of a rigid body. But the most remarkable illustration of 

 them is derived from the two different ways in which it is possible to contem- 

 plate the relation between electricity and magnetism. 



Helmholtz, in his great paper on Vortex Motion, has shewn how to construct 

 an analogy between electro-magnetic and hydro-kinetic phenomena, in which 

 magnetic force is represented by the velocity of the fluid, a species of translation, 

 while electric current is represented by the rotation of the elements of the fluid. 

 He does not propose this as an explanation of electro-magnetism ; for though 

 the analogy is perfect in form, the dynamics of the two systems are different. 



According to Ampere and all his followers, however, electric currents are 

 regarded as a species of translation, and magnetic force as depending on rotation. 

 I am constrained to agree with this view, because the electric current is 

 associated with electrolysis, and other undoubted instances of translation, while 

 magnetism is associated with the rotation of the plane of polarization of light, 

 which, as Thomson has shewn, involves actual motion of rotation. 



The Hamiltonian operator FV, applied to any vector function, converts it 

 from translation to rotation, or from rotation to translation, according to the kind 

 of vector to which it is applied. 



I shall conclude by proposing for the consideration of mathematicians certain 

 phrases to express the results of the Hamiltonian operator A. I should be 

 greatly obliged to anyone who can give me suggestions on this subject, as I 

 feel that the onomastic power is very faint in me, and that it can be success- 

 fully exercised only in societies. 



V is the operation ; _ +y _ + fc - , 



where i, j, k are unit vectors parallel to x, y, z respectively. The result of 



* The subject of linear equations in quaternions has been developed by Professor Tait, in several 

 communications to the Royal Society of Edinburgh. 



