148 PETER GUTHRIE TAIT 



tion of energy, & therefore a tendency to set like a magnet. The comfort is that 

 V'o- cannot subsist of itself. 



" Of course the resultant force on an element is of the form FVV, and if <r is 

 a function of z only, and Ska- = o, 



"This is the only explanation of terms of this form in an isotropic or fluid 

 medium, and since the rotation of plane of polarization is roughly proportional to 

 the inverse square of the wave length, terms of this form must exist. 



dp,, 

 df 



Thus, on the one hand, we have Tait submitting his quaternionic 

 theorems to Maxwell's critical judgment, and Maxwell recognising the power 

 of the quaternion calculus as handled by Tait in getting at the heart of 

 a physical problem. 



Unfortunately Tail's letters to Maxwell have not been preserved ; and 

 we can only infer as to the general nature of Tail's replies to Maxwell's 

 constant enquiries regarding quaternion terms and principles. There can be 

 no doubt however that, in introducing the operator V and the Hamiltonian 

 notation associated with it, Maxwell was strongly influenced not only by 

 Tail's masterly paper on Green's and other Allied Theorems but also by 

 his intimale correspondence. 



The fundamental properties of V as a differential operator may be 

 expressed very simply in dynamical language. When it acts on a scalar 

 function of the position of a point it gives in direction and magnilude the 

 maximum space rate of change of this funclion. For example, if u is a 

 polenlial, V is the corresponding force. Its effect upon a vector quantity 

 is, in general, to produce a quaternion, with its scalar and vector parts. 

 Suppose the vector quantity to be the velocity of flow of a fluid, symbolised 

 by cr ; then Vcr consists of two parts, the scalar and vector parts. The former, 

 SVcr, represents what Maxwell called the Convergence, indicating a change 

 of density in the fluid at the point where cr is the velocity ; and for the 

 latter, symbolised by FVcr, and measuring in the present case the vorticity, 

 Maxwell's name of Curl has been generally accepted. 



It is instructive to read Tail's early papers discussing the properties of V, 

 and to follow the growth of his power in dealing wilh it. At first he was 

 contenl lo begin wilh Hamillon's trinomial definition, as in the paper of 1862 



