1885.] equations of vibrations of ether in a light wave. 281 

 1. The equations of which the solution is required are 



. 5-w «• 



and similar equations for 77 and £", where y 2 stands for the Cartesian 



fd 2 d 2 d 2 \ 

 operator ^+.^+^j. 



After solution I shall add the condition that 



d%_ + dv + <% = 

 dx dy dz 



I shall suppose that the time enters the solution only through 

 trigonometrical terms, and (in the first instance) that the disturbance 

 arises from a single source at the origin. In this case the solution 

 takes the form 



£ = 2 {A x sinp (r — at) + B x cosp (r — at)}, 

 where P = ^ r> ^ being the wave length, and generally our unit 



A, 



for comparison of magnitudes. 



I shall shew how A and B are to be expanded in series of 



terms descending in magnitude comparably with those of the 



\ \ 2 

 geometrical series 1 . - . -, . . . For on substituting this value of £ 



in the differential equation, we get these equations of condition : 



Write now A and B in a series, in descending order, of 

 homogeneous terms, beginning with one of degree — 1, to ensure 

 convergency, and use Euler's Theorem. Thus 



A x = u_ x + u_ 2 + u_ 3 + &c. 

 B 1 = v_ % + v_ 2 + v_ z + &c. 



••• V 2 fa* + «-. + &c.} + ^ {w_ 2 + 2v_ 3 + &c.} = 0, 



V 2 K + V -2 + &C 'l - % i U -2 + 2W ~3 + &C '} = 0. 



(2). 



