280 Discussion of a New Theorem in Wave Propagation. 



other kinds, the wave-surface round a centre of disturbance 

 is no longer a sphere or spheres : it is a wave-surface of some 

 other form. If it be a surface which when referred to 

 rectangular coordinates has an equation of which the para- 

 meters are of the form at, bt, &c, where a, b, &c. are constant 

 velocities, then when referred to polar coordinates the velocity 

 of propagation in the various directions is a function of 

 and (j) only, and remains unchanged in each direction. Ac- 

 cordingly, as the wave advances and the surface enlarges, its 

 form remains the same. Hence when these wave-surfaces 

 are drawn round the pun eta p, p', &c. of the diagram on 

 p. 276, their common tangent-plane need no longer be per- 

 pendicular to the axis of the cylinder, but will, in general, 

 be oblique to it. It appears, then, that the only change that 

 needs to be made is to allow the plane AB to assume in the 

 diagram any position, whether perpendicular or oblique to 

 the axis of the cylinder. The rest of the proof then proceeds 

 as before, so that the theorem is true in all media in which 

 the wave-surface belongs to the family of surfaces described 

 above. This includes all known uniform media. 



If we could in any experiment isolate one of the undula- 

 tions of plane waves we should see it extending indefinitely 

 sideways, i. e. in the direction of the plane of the waves. But 

 the most we can effect experimentally is to isolate small 

 sheafs of these undulations, of which the axial rays (i. e. per- 

 pendiculars from the origin on the planes of the waves) lie 

 within a small cone ; as in the experiment described in § 42 

 of the paper on Microscopic Vision in the December Number 

 of the Philosophical Magazine, p. 525. In that experiment 

 we find that if two such sheafs of undulations are allowed to 

 interfere and produce a ruling, this ruling will be seen to 

 extend more and more laterally the smaller we make the 

 sheafs ; and they would extend laterally to an indefinite ex- 

 tent if we could indefinitely decrease the angle of the cones 

 within which lie the axial rays of the undulations which 

 make up the sheafs. 



The method of investigation followed in the preceding 

 pages enables us to acquire a singularly clear insight into how 

 it comes to pass that a disturbance, however complex, can be 

 resolved into undulations of absolutely uniform plane waves. 

 Of course it follows as an easy corollary that resolutions into 

 undulations of curved waves are also possible ; but none of 

 these has the tw T o supreme advantages of consisting of waves 

 which are uniform over the whole of each wave-front, and 

 which when they advance to distances small or great undergo 

 no change either in intensity or in any other respect. 



