278 Dr. G. J. Stoney on a New Theorem 



existed on the outward journey, except that the motions are 

 now reversed. This conception makes it specially easy to 

 picture the undulations of plane waves as flowing in from all 

 directions and across the space that had been occupied by the 

 original disturbance, and while crossing it reproducing in it 

 the original disturbance with all its motions reversed. We 

 may, if we choose, complete the picture by imagining another 

 reversal of all the motions afterwards to take place, when the 

 undulations will again cross the situation of the original 

 disturbance, reproducing it as it originally existed and with- 

 out any reversal of its motions. 



A very simple contrivance will here enable us to realise 

 more fully what it is that occurs in nature. To this end let 

 us imagine a sphere to be described including the space 

 throughout which we have supposed the original disturbance 

 to prevail, and across and beyond which we now know that 

 undulations of uniform plane waves of indefinite extent are 

 advancing in all or some directions. This sphere may include 

 also any portion of the surrounding space of which we may 

 desire to investigate the motion, Next imagine cylinders 

 enveloping this sphere to be pointed in all the directions in 

 which the undulations advance. Each cylinder will then cut 

 what we may call a beam out of the undulation proceed- 

 ing in its direction, and the whole of each undulation may 

 be distinguished into the two parts— the beam or part inside 

 the cylinder, and the rest of the undulation which lies out- 

 side it. Now after the reversal it is only these beams that 

 can reach any part of the space inside the sphere ; they are 

 therefore the only parts of the undulations with which we 

 need concern ourselves if our only object is to investigate 

 the events which occur within the sphere. We do not need 

 to take into account the cooperation of other parts of the 

 undulations unless we want to investigate events that he 

 "beyond the sphere. We may call the beams the effective 

 part of the whole system of undulations, and the rest of the 

 undulations the inoperative part with respect to events within 

 that sphere. 



It will accordingly suffice to think of the sphere as emit- 

 ting these beams on their outward journey, and again to 

 picture to ourselves these beams after the reversal as con- 

 verging in upon the space within the sphere*. 



* The only effect of the rest of the undulations upon these beams is 

 to prevent the formation along their bounding cylinders of those special 

 conditions which are met with over the sides of ordinary beams. The 

 plane waves of which they consist remain uniform quite up to their 

 bounding cylinders. 



