378 
Proceedings of Royal Society of Edinburgh. [sess. 
§ 19. Two kinds of incident solitary wave are expressed by (41), 
of types represented respectively by the following elementary 
algebraic formulas : — 
t-ax + by 
(t-ax + by ) 2 + r 2 
and 
r 
(t - ax + by) 2 + r 2 
(43). 
The same formulas represent real types of condensational waves 
with £la and rjl (- c ); instead of the £/b and y/a of (41) which 
relates to distortional waves. It is interesting to examine each of 
these types and illustrate it by graphical construction : and par- 
ticularly 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 4- by) 2 which is 
large in comparison with r 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 & /} or c, or c t is imaginary. They are as 
follows, for example if b t = lq, where g is real ; 
t — ax 
( t-ax) 2 + (gy + r ) 2 
and 
(«), 
gy+l 
(l - axf + (gy + r)- 
These real resultants of imaginary waves are not plane waves. 
They are forced linear waves sweeping the interface, on which 
they travel with velocity a -1 ; and they produce disturbances 
penetrating to but small distances into the medium to which they 
belong. Their interpretation in connection with total internal 
reflexion, both for vibrations in the plane of the rays, and for 
the simpler case of vibrations perpendicular to this plane (for 
which there is essentially no condensational wave), constitutes the 
dynamical theory of Fresnel’s rhomb for solitary waves. 
