414 Mr. O'BRIEN, ON THE PROPAGATION OF LUMINOUS WAVES 



$ 18. To shew that the condition of the vibrations being transversal is 



da dfi dy _ 

 dx dy d% 



Let the equation to any surface in which all the particles are in the 

 same phase of vibration (i. e. any wave-surface) be 



Fix, y, &,) = u, 



where it is a parameter which does not vary as long as x, y, as belong 

 to the same wave-surface, but is different for different wave-surfaces : 

 for example, if the wave-surface be spherical we may take u to represent 

 the radius, or if it be plane we may take u to represent the perpendicular 

 upon it from the origin. In the former case the above equation would 

 be 



& + "if + z * — u "> 



and in the latter 



px + qy + s% = u, 



where p,. q, s, represent the cosines of the angles which the surface 

 makes with the co-ordinate planes. It is evident that in both these cases 

 u varies only when we pass from one wave-surface to another. 



Now supposing that t is constant, the phase of vibration, or what 

 is the same thing, a, /3, 7 can vary only when u varies, hence a, (Z, y 

 must be functions of u and t only : moreover, we may suppose u and t 

 to alter in such a manner that the phase of vibration shall not alter, that 

 is* we may suppose that du and dt are so taken that da, dfi, dy are 

 each zero; hence, supposing du and dt thus taken, we have (remembering 

 that a, /3, 7 are functions of u and t alone). 



da ,. da 1 „ ,_■„ 



-r-dt + j- du = ... (1). 

 dt du 



^* + g*.o... <*>. 



%Jf '+ £W - ... (3). 



