TRANSACTIONS OF SECTION A. 573 



and accurately uniform, and at the same time to increase the size of sphere r 

 to any desired extent. When this has been done we obtain the following im- 

 portant theorem : — 



Theorem I. 



The undulation of spherical waves emitted by a luminous punctum P situated 

 at a point / of a transparent isotropic medium, together with that preceding 

 system of waves converging upon /, which would have been followed by this 

 same radiation from / if P had been absent — i.e., the complete undulation of 

 spherical waves which embraces an entire past history as well as the entire 

 future history of the undulation — can be completely resolved into undulations 

 of plane wavelets, each wavelet being of unlimited extent in its own plane, and 

 uniform throughout that extent. And this resolution is a true physical resolution 

 and not merely kinematical. 



An adequate conception of these plane-wavelet components can perhaps be 

 best acquired by making temporary use of the hypothesis that the light emitted 

 by P consists of rays, of the kind with which we are familiar when the useful 

 hypothesis that light consists of rays is made the basis of the science of 

 geometrical optics. Here, however, we are to obliterate these hypothetical rays 

 and to substitute for each hypothetical ray a real undulation of plane wavelets, 

 each wavelet having its wave-front perpendicular to the ray, and being of 

 unlimited extent in the plane of the wavelet as well as uniform throughout that 

 extent. To complete the picture the intensity of each imdulation {i.e., the 

 square of the transversal of each of its wavelets) is to be proportional to the 

 intensity which we have to attribute to the corresponding hypothetical ray of 

 geometrical optics. As the number of rays is unlimited, so is the number of the 

 undulations of plane wavelets that take their place. 



The investigation requires one other fundamental thereom, of which, as it is 

 a well-known theorem, we need only give the enunciation, premising that the 

 direction in which an undulation of plane wave.s travels is in an isotropic medium 

 perpendicular to the wave fronts. 



Theokem II. 



Any number of undulations of uniform plane waves, of wave length X, 

 advancing in the same direction in an isotropic medium, may be united into a 

 single resultant undulation of uniform plane waves travelling in that direction. 

 (If the undulations to be combined are variously polarised, the resultant imdu- 

 lation will in general be elliptically polarised.) 



From these fundamental theorems several useful inferences may be drawn ; 

 such as — 



Theoeem III. 



The whole of the light of wave length X emitted by any visible object, 

 •whether self-luminous or requiring incident light to render it visible, may be 

 resolved into undulations of uniform plane wavelets, of which there need be only 

 one such undulation provided for each direction towards which light is propa- 

 gated from the visible object. 



This is an immediate corollary from Theorems I. and II. 



Theoeem IV. 



The light of wave length X traversing any portion of space may be resolved 

 into undulations of uniform plane wavelets sweeping over that space, of which 

 there needs only one such undulation in each direction. 



This also is R corollary upon Theorems I. and TI, 



