190 On Reflexion and Refraction of Solitary Plane Waves. 

 §18. Putting now 



I=p + iq (40); 



and from this finding V, I n J', J / ; and taking for the real 

 incident wave-motion (§10 above) 



f = V _ i r p + iq p-ig -| "| 



b a 2 Lt—ax + by + iT t — ax + by — trj \ 



_ p(t — ax + by) + qT | 



(t-ax + by)* + T 2 J 



being the mean of the formulas for -f 1 and —t; we find a 

 real solution for any case of b n c, c n some or all of them 

 imaginary. 



§19. Two kinds of incident solitary wave are expressed by 

 (41), of types represented respectively by the following 

 elementary algebraic formulas : — 



t-ax + by 



and 



{t—ax + byf + i* 



(t-ax + byf + T* 



(43). 



The same formulas represent real types of condensational 

 waves with f/a and y/( — c), instead of the f/6 and rj/a of (41) 

 which relates to distortional waves. It is interesting to 

 examine each of these types and illustrate it by graphical 

 construction : and particularly to enquire into the distribution 

 of energy, kinetic and potential, for different times and places 

 in a wave. Without going into details we see immediately 

 that both kinetic and potential energy are very small for any 

 value of (t—ax + by) 2 which is large in comparison with t 2 . 

 I intend to return to the subject in a communication 

 regarding the diffraction of solitary waves, which I hope to 

 make at a future meeting. 



§20. It is also very interesting to examine the type- 

 formulas for disturbance in either medium derived from (41) 

 for reflected or refracted waves when b n or c, or c/ is 

 imaginary. They are as follows, for example if b ; = ty, where 

 g is real ; 



t ^ ax ... . (44) 



(t-ax) 2 + (gy + T) 2 { h 



9y + T ... (45). 



(t-ax)*+(gy + r) 



