in Wave Propagation. 211 



the theorem in section 38, p. 435 of the Philosophical Maga- 

 zine for last December, combine into a single plane wave in 

 that position. 



Again, spherical waves round p, p', &c. are not uniform, 

 but the departure from uniformity, since it depends on the 

 position of the transversal, varies gradually over each spherical 

 wave ; and when the radii r, r' ', &c. are large quantities of the 

 first order, this want of uniformity is only a small quantity of 

 the first order over the sectors cut off by the cylinder. Hence 

 the individual waves and therefore the resultant wave are 

 uniform within those limits. 



Hence, finally, the disturbance proceeding in the direction 

 of the axis of the cylinder becomes ultimately an undulation 

 of uniform plane waves. 



From the proof it appears that the lateral extent which 

 may legitimately be given to the cylinder KQ, and therefore 

 to its sections QR and LK, increases, and increases without 

 limit, when the distance of the plane AB from the puncta is 

 increased without limit. Hence the disturbance proceeding in 

 the direction pb becomes at the limit an undulation of uniform 

 plane waves, each of unrestricted extent in the plane of the 

 wave, i. e. perpendicular to the direction pb. It is in fact 

 one of the waves included in the summation 



pfMsin ^ + ^ + ^-^ +tt)] .da>. 



This being true whatever direction be taken for pb, the pro- 

 position is proved in the case we have been considering, 

 i. e. in the case of monotropic media. 



. It is very instructive to conceive all the motions reversed 

 at a given instant, so that the plane waves of infinite extent 

 travel inwards *, reproducing at each stage of their inward 

 progress the same disturbed state of the medium as had 



propagate undulations backwards, which as they reach any other 

 section of the cylinder, LK, however distant, will still differ in phase 

 from one another only by infinitesimals, i. e. by quantities which it is 

 legitimate to disregard, and will differ in no other respect except by 

 quantities of a still higher order of small quantities. 



* In carrying out this conception it should be borne in mind that, in 

 a monotropic medium, each part of an infinite undulation of uniform 

 plane waves simply advances without change in the direction perpendicular 

 to the plane of the waves. If the medium be not monotropic, the undu- 

 lation as a whole still advances without loss of intensity perpendicularly 

 to the plane of the waves ; but this is now by reason of each part of it 

 advancing without change in an oblique direction. 



