of Maxwell's Electrical Theory, 181 



are functions of z and t only, which are perpendicular to k 

 [i,j, k taken as usual with reference to the rectangular axes 

 #, y, z] . Thus 



v'=*a/a*, (i9) 



and therefore y' is commutative with k. Put 



E -kH =2x, E o +M =2& . . . (20) 



which give 



E = « + A H = *(*-/3) (21) 



Equation (18) now gives 



S(*B*/as+£d/3/B2)=0, 

 or 



a* + /3 2 = -X 2 , (22) 



where X is a function of the time only. 



From the constancy of the Poynting intensity we see that 

 it is needless to consider a more general value of a' than uk, 

 where wis a scalar function of z and t only. Substituting 

 this value and multiplying the second of equations (17) by k 

 and adding it to and subtracting it from the first we obtain 



aa/d< + 3{(w + v)*}/3* =0 . . . . (23) 



Wftt + -di(u-v)P 3* = . . . . (24) 

 These give at once 



a=B7/d*j(M + «0*= -3y/B*, • • (25) 



/3=-d8fdz,(u-v}P= --dBftt, . . (26) 



where of course 7, S, like a, f$ 9 are vector functions of z and t 

 only, which are perpendicular to k. 



Suppose y= r Yii + r Y2J, where 7 X and <y 2 ar e scalars. Since 

 "dyfoz and 'dy/'dt are parallel, it follows that 



tz I ~dt ~dz J & 

 or 



. j (r;)=° ( 27 > 



Hence 7x and % and therefore y are functions of a single 

 scalar function 6 of z and £. 6 is to a certain extent arbitrary, 

 since any function of will serve instead of 0. Thus 



7=7(0), (28) 



