Electromagnetic Theory of Light. 235 



that direction, we shall have 



H (or E &c.) = function (z — vt), 

 and for any such function 



Thus (1) becomes 



VA'D ,H = - z;DJ), YkDX = vD z B ; 



or putting, in accordance with the definition of a as an index- 

 vector 



kv~ 1 = <r, 



D=-Vo-H, B=WE (4) 



By (4) and (3), 



SDE = S<7EH = SBH=-w (5) 



By (4) and (5), 



YDB/w=<r. (6) 



So far we have been practically following Mr. Heaviside. 



By (2), 



SD^7E = SE</D, SBdH = SH<7B. 



Hence by (5) 



-±<7i0 = SDdE = SEc/D = SBrfH=SHdB. . . (7) 



Now define p by the equation 



VEH/«? = /a t $) 



By (5), 



So/>=-l . , (9) 



By (4), 



SHdB = SHdV o-E = SHo-dE + Srfo-VEH 

 = SDcZE + wS/od<r. 



Hence by (7) and (9), 



S/>do-=0, So-<tfp = (10) 



This shows that the locus of the extremity of p and that of 

 a are polar reciprocals, and therefore that p is the vector of 

 ray velocity. 



Putting now, in accordance with our first definitions, 



P=oVu>, B=6VWj B = /3v/w, H = 7 ^w, . (11) 



