402 The Astronomer Royal on a difficulty 



Now, as the first subject of my communication, I propose to 

 compare this with the equation applying to the motion of long 

 waves of water in shallow canals, when the proportion of the 

 vertical movement of the particles to the depth of the canal 

 is not neglected ; a case which I have in some measure dis- 

 cussed in the Encyclopaedia Met7-opoUtanaf article Tides and 

 Waves. That equation has the form 



2^_/ dxy d^X 



The equations are generally similar in form, with the differ- 

 ence that the exponent in one case is 2 and in the other is 3. 

 Imperfect solutions of both (as my friend Professor De Mor- 

 gan has pointed out to me) may be obtained in the following 

 forms : — 

 For the first. 



For the second," 



dX ^ , / dX\ 



dX ^ 2a 



-df=^-^ 



x/l-f}" 



These solutions give us no assistance further than this, that 

 they indicate a critical state of the movement of the particles 



J'SJ' 



in both as occurring when -t-= —1, which in both equations 



makes — r— infinite. 

 dt 



But a very distinct idea of the nature of the progressing 



wave in both cases may be obtained by solving the equations 



by the method of successive substitution. Thus, of the equation 



rf^ _ ^ _ r^ (dxyi cpx 



' ds^ dt'' ~ V ^^ \da;J J * dt^* 

 a first solution may be obtained by neglecting the right-hand 

 side (a quantity of the second order as regards the movement 

 of the particles), which solution, if the wave be supposed to 

 travel in one direction only, is 



X = c.f{at-,v). 



If this value be substituted in the right-hand side of the equa- 

 tion, terms of the second order only being retained, and if the 

 solution be then effected under the conditions proper for suc- 

 cessive substitution, namely that no new terms of the first 

 order be introduced, but that the terms of the second order 

 may have the most general form ; and if we avail ourselves 



