344 REPORT — 1893. 



theory, the energy of this medium is made up of the kinetic or electro- 

 magnetic part T, and the static part W, where in Maxwell's notation, dr 

 being an element of volume, 



T=i[(aa + 6/3 + cy)c?r, 

 W=||(P/+Q^+R;Odr; 



and there is also in our problem to be added on another small term, 

 Maxwell's hypothetical magneto-optic part T'. 



Now the dynamical equations of any medium or system are most 

 fundamentally expressed as the conditions that the characteristic function 

 of Lagrange and Hamilton 



should be stationary for a given time of motion from any one definite 

 configuration to another, subject to whatever restraints the coordinates 

 have to obey. This form is the most fundamental, because the processes 

 of the Calculus of Variations are purely analytical, and quite independent 

 of whatever specifying quantities we may choose in order to represent the 

 state of the system, the only condition being that the function T + T'— W 

 is to be expressed in terms of a sufficient number of measures of configura- 

 tion and their first diSerential coefficients with respect to the time, and 

 is to be of the second degree as regards these differential coefficients. 



In order to obtain such an expression Fitz Gerald proposes to treat 

 (a, /3, y) as velocities corresponding to coordinates (I, j/j C), so that 



and then 



(«,/3,y)=|(sS»/,0, 





and this will be successful if W can be represented in terms of {I, rj, ^) 

 only. Now in a dielectric 



dt dy dz^ ' ' • » 

 hence 



also (P, Q, E,) is, from the constitution of the medium, expressed in terms 

 of {f, g, h) by the linear equations of electrostatic induction, so that the 

 thing required is done. If, in fact, 



W= f XJdT, 



where U is a quadratic function of (/, g, h), the equations of motion in 

 non-rotational media are involved in the variational equation 



