1885.] equations of vibrations of ether in a light wave. 285 

 It will be sufficient to prove the first, 



dx iK ~ \dx) + ^dx*' 

 using this and the analogous relations 



«W + V + O = W {(I)' + (|) ! + g) 8 + f V'l + etc. 



By the consideration of the dimensions it is plain that this 

 equation is not accurate, since in that case the intensity would be 

 expanded in a series similar to that for £, &c, in a manner cor- 

 responding to that used for x% + yn + z£. 



2. The order of magnitudes which we are considering com- 

 pared with the amplitude of the disturbance, which we shall call 

 simply d, is represented by 



f. dP d% . /h7X 



? : dt :a 7r X:a]a (7) - 



and therefore 



df • \dt) dxdt ' \dx) dx 2 ? • ? w> 



Hence, in neglecting terms in the equations of motion of an elastic 

 solid which contain squares and products of the differential co- 

 efficients, we assume that £/\ may be neglected while our only 

 declared rule has been to neglect \/r; which is really treating 

 them as comparable quantities. (With regard to this ratio f/A,, see 

 Sir W. Thomson's paper quoted above, where it may be taken as 

 not greater than 1 : 300.) In what follows I shall suppose that 

 we may retain £/A but neglect the square. 



We have thus to extend our equations by including terms 

 such as appear in the hydrodynamical equations and in the 

 general theory of elasticity. If, in fact, a theory of molecular 

 vortices, such as is touched upon by Maxwell [Electricity and 

 Magnetism, Vol. II. Art. 823), is to be considered at all, the terms 

 of the second degree upon which the theorems about vortices 

 depend must be retained in the equations. I shall prove that the 

 expressions already found for £, tj, f identically satisfy the complete 

 differential equation, a result which would be necessary if we are 

 not to introduce terms affecting waves of half the length or say 

 octaves of the original wave length. 



vol. v. PT. iv. 20 



