in Wave Propagation. 275 



If the problem be merely kinematical the properties of the 

 medium in which the motion takes place need not be con- 

 sidered, since the form of wave-surface round a point, the law 

 for the propagation of waves through the space and the legi- 

 timacy of the geometrical superposition of motions within it 

 may all, in a kinematical investigation, be arbitrarily assumed. 

 But the case is different when we are dealing with nature. 

 Here we must take into account the physical properties of 

 the medium — the form of the wave-surface round each centre 

 of wave propagation, the number of sheets it has, and the 

 character of the wave motion in each of these ; and we must 

 limit ourselves to the cases in which the geometrical super- 

 position of the motions is legitimate. 



We shall begin with the simplest case, that of a monotropic 

 medium. Here the wave-surface round any centre of dis- 

 turbance is one or more concentric spheres — usually two, one 

 belonging to the longitudinal the other to the transverse 

 waves. 



Again, an undulation of spherical waves of any kind emitted 

 from a centre of disturbance may be divided into a longitu- 

 tudinal and two transverse trains of waves along each radius 

 with transversals at right angles to one another, if the motion 

 be a dynamical one ; or into whatever correspond to these if 

 the undulation be of any other kind ; and each of these may 

 by Fourier's theorem be regarded as due to the coexistence of 

 trains of waves of the simple pendulous kind. We accordingly 

 need only concern ourselves with these last. 



Let then p, p f , &c. be puncta, i. e. centres of disturbance, 

 in such a medium. Draw any plane AB at a distance from 

 them and perpendiculars r, r', &c. from the puncta to this 

 plane. Also draw a cylinder KQ with one of these, say r, 

 as its axis, and with a radius k large enough to include within 

 the cylinder all the puncta. Then draw spheres round the 

 individual puncta, touching the plane AB at b, b r , &c. to 

 represent the waves which had emanated from the puncta* 

 and which have reached the plane AB at a given epoch T. 



* The coEtribution from each punctum is to be estimated as that due 

 to the difference between its motion, and the motion which would have 

 reached its situation from the operation of all the other puncta after 

 making a similar allowance in their case. See on this subject a " Note 

 on the Propagation of Waves "in the Transactions of the Royal Irish 

 Academy for 1860, p. 37, in which what is here called the contribution 

 from a punctum is there called its influence. 



The necessity for estimating the contributions in this way has been 

 much overlooked in textbooks on Light, &c, where the subject is usually 

 so presented as to lead to erroneous conclusions, such as that a wave 

 ought to propagate itself backwards as well as forwards. 



Y2 



