F= 



1884.] of the electromagnetic field. 145 



Again substituting in (5) the value of X which we have found 

 in (9), we obtain as the value of F, if we omit the terms in V, 



H \\ I - dx'dy'dz 



-i/Z/S^S)^*; ^ 



Let us consider a plane wave of magnetic force travelling 

 through the medium in a direction whose direction cosines are 

 I, m, n. Let %, Jill, "N be the direction cosines of the magnetic 



force, then we may put 



2tt 



a = A% sin -—(lx + my + nz - Vt) (18). 



A. 



etc 



Since ^+f + ^ =0 , 



ax ay dz 



the lines whose direction cosines are I, m, n, %, J^l, ftT respectively, 

 are at right angles, that is, the magnetic force is in the plane of 

 the wave. 



Now let L, M, N be the direction cosines of any line normal 

 to %, J$H, $T, and e be the angle between I, m, n and L, M, N, 

 and put 



9 



— - (Ix + my + nz — Vt) = 8. 



A, 



mi ~ ( mN—nM 



lhen 1L = : , etc. 



sine 



.(19). 



, . (mN-nM) . _, 1 fdH dG\ 



and a = A = - sin 8 = - -, r - 



sin e /a \ ay dz J 



etc 



These are satisfied by 



r, ulAL\ -, d® /ewVk 



F = - f T . cosS + -j- (20), 



Z7T sm e a# 



etc 



® being any function of x, y, z and t. 



* ■ dy d/3 A d 2 <& 



Again, since —- — f- = 47rw — 



eft/ cfe dxdt' 



4<7ru — , , = ^4 -— : — {w (Of— ??ii) - n (nL — IN)} cos 8 

 uxu/t A. sin e 



= - -=^- {£ (J 2 + m 2 + ?i 2 ) -l(Ll + mM+ nN)} cos 5 . . .(21). 



A. S1U 6 



etc 



vol. v. PT. II. 10 



