284 Mr R. F. Owyther, On the solution of the [May 25, 



(on the condition that -~ + -^ + -^ = 0) 

 dx ay az 



Hence x% + yrj + z% satisfies the fundamental equation of this 

 paper and can be expanded in the same form as £, 77, £ are them- 

 selves expanded, but the terms of order — 1 in £, 77, £ will appear 

 as of order in xg + yrj + z%, and therefore the sum of such terms 

 will vanish. We may use a convenient notation by writing this 

 x fi_ x + y v u_ x + z $i_ x = 0] 



xtv-t + y^ +z< ; v_ 1 = o\ 



That x% + yrj + z£ may vanish absolutely will require only that the 

 terms of order — 2 should vanish, or that 



« ; ^_ 2 +y v v_ 2 +z i v_ 2 = 0\ 

 These conditions reduce to 



d d d . 



Tx^ + dy^ + Tz< U -i = ° 



d d d 



d^^ + dy^ + dz^ = ° 



(5). 



In ordinary cases of the propagation of light waves, we may 

 neglect all parts of ff, rj, £ except the greatest, and in finding the 

 differential coefficients we need only to differentiate the trigono- 

 metrical function. 



In this case we have 

 d% _x^d% d% _ y d% d% _ z d% d% _ 1 d% ,,,, 



dx ra di' dy ra dt' dz~ ra~dt' dr a dt" 

 and the variation of the displacement in directions perpendicular 

 to the wave normal, or in the wave front, i.e. the elongation in lines 

 in the wave front, is zero. This indicates that the components 

 of the strain are greatest when the velocity is greatest, and that 

 at any point the kinetic and potential energies have their zero and 

 maximum values simultaneously. 



By the relations already established, simple proofs can be 

 given that the kinetic energy and the intensity (measured by 

 f 2 + rf + O of the disturbance are propagated by the same law as 

 the disturbance itself except near the origin of light. That is that 



and S-«v|{f + ') 2 +n = 0. 



