of a Theorem in Wave-motion, 371 



such 



as 



v=M*> &>*>*)> 



(where £, 77, and ? may be displacements, or velocities, or so 

 on, in the three coordinate directions) ; and by then expand- 

 ing each of these by Mr. Preston's theorem. But on a close 

 scrutiny we find that though this furnishes an apparent solu- 

 tion, in the form of forced vibrations, or rather a group of 

 such solutions, this group unfortunately does not include the 

 solution which would be selected by nature under any con- 

 ceivable circumstances. The analysis furnishes undulations 

 which could not propagate themselves through any medium. 

 The motions which it furnishes are the non-natural motions 

 of a mere forced kinematical resolution, of no use in physics. 

 That this is so can be made plain by taking any very simple 

 example, such as the following. 



Let the " given region of space " within which the disturb- 

 ance is maintained be a thin circular disk perpendicular to 

 the axis of z ; let the origin of coordinates be at the centre 

 of the disk, let the disturbance maintained within it be of 

 the simple kind represented by 



£=f {vt -z)+f(vt + z), 



and let the medium be the sether. 



The undulations which will be generated by this disturbance 

 will propagate themselves both forwards and backwards and 

 both within and beyond the disk ; and as from symmetry the 

 two groups of undulations will be exactly alike, it will suffice 

 to ascertain what those travelling forwards will be. This is 

 easily done from the circumstance that they closely approxi- 

 mate to being identical with the radiations forwards from a 

 circular opening in a screen of the size of the disk, when the 

 lio-ht from a star, or rather that part of it polarized in one 

 direction, is allowed to fall perpendicularly on the back of the 

 screen. 



Before reaching the screen the light from the star of any 

 one wave-length is as near an approximation as can be realised 

 to being a single undulation of uniform plane waves with 

 each wave of infinite extent in its own plane. A cylindrical 

 beam out of this undulation is what reaches the opening in 

 the screen. Until it reaches the opening it is an absolutely 

 single train of uniform plane waves. But at the opening it 

 ceases to be this single beam. From that situation forwards 

 it spreads in a highly complex way over what we may call a 



