276 



Dr. G. J. Stoney on a New Theorem 



The times at which they had started from their respective 

 centres may be represented by T— ■*, T — 1' 9 &c. Let now 



^E=z 



the plane AB be removed so far from the puncta that the 

 radii r, r', &c, become large quantities of the first order ; 

 and consider the sectors of the spherical waves which are cut 

 off by the cylinder KQ. The sagittas of the spherical waves, 

 i. e. the intervals between the spheres and their common 

 tangent-plane, nowhere exceed small quantities of the first 

 order over these sectors : and accordingly plane waves lying 

 in the plane AB may he substituted for them*. Now these by 



* The legitimacy of this substitution is usually assumed. It may "be 

 proved as follows : — It arises from the circumstance that in the propaga- 

 tion of waves which are approximately plane and uniform to distances 

 however great, differences of phase which were infinitesimal at starting 

 remain infinitesimal to whatever distance they may be propagated ; and 

 that changes of phase which are infinitesimals of the first order can only 

 involve changes of amplitude of the second order of small quantities. 



Hence if all motions were at a given instant reversed in one of the 

 sectors of spherical waves cut off by the cylinder— suppose in that which 

 emanated from p, — and simultaneously reversed in the wave we have 

 substituted for it in the plane QR (viz. in that part of the tangent-plane 

 cut off by the cylinder) : then the corresponding elements of these 

 differ in phase only by infinitesimals and differ in no other respect. 

 For simplicity suppose the medium otherwise undisturbed. Then these 



