372 Mr. Earnshaw on the Motion of Luminous Waves. 



dYm _dY{R) d^YjR) d^Y{R) , 



dx - '^dx + rf"^2 " iS - U + -^^^r^ 11 iJ 



+ ?li^'«-"- 



Hence 



Now the first term of the right-hand member of this equation 

 is zero, because it refers to the equilibrium position of the 

 medium. With respect to the next term, we observe that 

 ^ — ^' has the same value for every particle situated in the 

 same wave surface, though its value is not the same for pai'- 

 ticles which are in different wave surfaces, except those sur- 

 faces be distant from each other by an exact wave's length. 

 We see also that the particles of the wave surface in whicli m 

 is situated exert no influence upon m. Let us, then, setting 

 out from m and passing over a wave's length, number the 

 wave surfaces 1, 2, 3, in order. Denote by A the va- 

 lue of 2 ( m' — rr^i ) for all particles which 



are in the ?th wave surface, and in all other surfaces through 

 the whole medium distant from the rth by multiples of a wave's 

 length. Then 



the symbol 2 now referring to summation for all values of r. 

 I shall now assume the law of displacement at the time t to be 



l^-a sin (r h + T), 



T being an unknown function of t, and h a constant depend- 

 ing upon the nature of the medium. I am borne out in as- 

 suming this law by experiment; but if it be objected to as not 

 sufficiently general, it will be understood that what follows 

 applies only to media in which this law of disturbance can 

 be transmitted. From this equation we find ^ = a sin T ; 



. rh . 

 and therefore f — ? = 2 a sin* — sin T — « sin r h cos T 



^ Jl ^-^ 



= 2 sin^ -— . ^ — sin r // Va^ — 0^ ; and consequently 



