260 Mr. J. MoCowan on the 



superposing the forms of the two oppositely moving systems 

 of waves of which it may be regarded as the result. It is 

 clear, however, from (17) that the displacement of any plane 

 from its mean or undisturbed position is simply the sum of 

 the displacements due to the component waves. 



4. A General Solution : Solutions in Finite Terms. 



The solution which has just been discussed is the simplest 

 of a series of complete solutions in finite terms which may be 

 obtained for a corresponding series of forms of cross section. 

 For if we take <j>(z) = (a + bz) n , then taking advantage of (1), 

 (4) may be written 



d 2 £_ g{a + bh) d*% 

 dt 2 ~ rib dx 2 



d * ; . . . . (23) 



dx 



and if we change the independent variables in this to u — d^\dt 

 and p = d%/dx, and the dependent variable to ^—px-\-ut — f, 

 we get 



D , + i/.^- .y(«+M) '* 8 *. (2 ±) 



P df nb du*' ' ' ' ' K } 



an equation of a form which has received considerable atten- 

 tion, the equations of Biccati and Bessel being connected 

 with it. The general solution of (23) will then be given by 

 the following equations : — 



dx <f>(z)~\a + bz$ ' * ' " ' V) 



{a + bz) f a + bzyd$ 

 X ~ ~~nb~\a^bii] dz> ' ' W 



■■« = £ W 



fa + bhl 

 la + bz i 



+ ut-?; = d; . . . (28) 



where $ is the general solution of (24), which may be easily 

 shown to be, on substituting for^? its value in terms of z, 



3 = \ /i(w— \/±ng{a + bz)/b . cos <f>) sin 2w </> . d<j> 



+ \ /2O+ \S4np(a-tbz)/b . cos <£) sin 2w </> . d<f> . (29«) 



J 



when n is positive, and 



