Theory of Long Waves. 255 



generality by taking the upper sign only. The complete 

 solution of (5) for an advancing series of waves of any form 

 is therefore given by 



-»<~$vW < 6 > 



where z = ~F (x) when £ = 0. Here the ordinate of the surface 

 at any time t is given for the plane of particles whose abscissa 

 in their undisturbed position was x, and we may obtain from 

 this f, u, and p by means of equations (1), (2), and (3). As, 

 however, it is generally more convenient and shows the cha- 

 racter of the motion more clearly, to have all quantities given 

 in terms of the actual abscissa f at time t, rather than the 

 abscissa x corresponding to an anterior undisturbed condition, 

 we shall work out the results in this form. 



Using the letter F with various suffixes to denote a series 

 of functions related to F in ways which will be sufficiently 

 obvious from the transformations without other explanation, 

 (6) may be written 



and therefore by (1) 



00) 00) w 00) 0(A) V 0'O) 



where 



and kt is written in place of the arbitrary function of t which 

 would naturally appear on integration, for we easily find on 

 substitution in (4) that this function reduces to the form kt y 

 k being an absolute constant. Thus, finally, 



z = F{£-(</ff.+( z )-k)t}, .... (8) 



where z = F(%) is the equation to the wave-form when £ = 0. 

 Again, 



d£ d%dz ,- f . 



00) ^ " 00) ™ tw f « m - f (A) V W) 

 f = F 2 0)-^+^0) V<7, 



Also 



(2) ~ dx~~ dz dx ~ dz 00) V <70(~)* dt ' 



" dzdt " V <f>'(~)' 



$W _d£ _d£dz _ d%<j)(h) / 0O) rf4 



v < 



T2 



