418 Mb. O'BRIEN, ON THE PROPAGATION OF LUMINOUS WAVES 



hence the first of the equations (F) becomes 



1 d 2 a „ d"a m t „ N 

 m df du 2 m 



and similarly, — -rk = -o-r^ -Cp 



J m dt du* m 



1 d 2 y d 2 y 



in 



■(G). 



m df "* du 2 



In exactly the same manner the equations (F) may be adapted to 

 the case of plane waves. 



§ 23. To adapt the equations (E) to the case of spherical waves. 

 The equation to the wave-surface in this case will be 



x 2 + y 2 + z 2 = w 2 , 

 u being now the radius of the surface. 

 In this case, we have 



da _ da du _ da X d 2 a d 2 a X 2 da U 2 — X s 



dx du dx du u' dx 2 



du 2 u 2 du 



,..-,, d 2 a d 2 a y 2 da u 2 — y* 

 and similarly, ^ = 3- ^ - 



du 2 «" 



d 2 a _ d 2 a Z* 

 dz 2 = ~ du 2 u 2 



+ 



du w 



da U 2 - J8 a 



du u 3 



hence we have 



d 2 a d 2 a d 2 a d 2 a 2 da 1 d 2 (ua) 



dx 2 dy 2 dz 2 du 2 u du u du 2 ' 

 hence the first of the equations (F) may evidently be put in the form, 



m. 



1 d 2 (ua) n d 2 (ua) 

 = Jo 



m df 



du 2 



m 



Cua 



j • -i i 1 d 2 {u(S) ~d 2 {u$) m J „ Q 

 and similarly, - —^ = B -££* - — ' Cu0 



m df 



du 2 



(H)- 



1^ d^ity) = _ d 2 (uy) _ m, Cu 

 m df ~ du 2 m 



* From these equations, if we obtain a a in the form f(u,t), we have a = -f{u,t),f{u,t), 

 being evidently the value of a for plane waves ; let a be the maximum value of f(u, t), then 



