1898-99.] Lord Kelvin on Reflexion and Refraction. 
6 1 1 
greatest of the wave-velocities, each of b, b /t c, c t is essentially real. 
A case of this character is represented by fig. 2, in which the 
velocities of the condensational waves in both mediums are much 
smaller than the velocity of the refracted distortional wave, and 
this is less than that of the incident wave which is distortional. 
When one or more of b , b j} c, c t is imaginary, our solution (26) (28) 
remains valid, but is not applicable to / regarded as an arbitrary 
function ; because although f(t) may be arbitrarily given for 
every real value of t , we cannot from that determine the real 
values of 
f(t + Lq)+f(t-Lq) (3"), 
and 
m+n)-f{t-^L)} (38). 
The primary object of the present communication was to treat 
this case in a manner suitable for a single incident solitary wave 
whether condensational or distortional ; instead of in the manner 
initiated by Green and adopted by all subsequent writers, in which 
the realised results are immediately applicable only to cases in which 
the incident wave-motion consists of an endless train of simple 
harmonic waves. Instead, therefore, of making / an exponential 
function as Green made it, I take 
/(*)= 
i 
t 4- IT 
(39), 
where r denotes an interval of time, small or large, taking the 
place of the “ transit-time ” (§ 7 above), which we had for the case 
of a solitary wave-motion starting from rest, and coming to rest 
again for any one point of the medium after an interval of time 
which we denoted by r. 
§ 18. Putting now 
I =zp + L q (40); 
and from this finding I', I /3 J', J / ; and taking for the real incident 
wave-motion (§10 above) 
f_ = V_ 
b a 
P + V + ff-tg 1 1 
t - ax + by + lt t - ax + by — itJ > 
pit - ax + by) + qT i 
(t - ax 4- by ) 2 + J 
(41) 
being the mean of the formulas for + 1 and -i; we find a real 
solution for any case of b t , c, some or all of them imaginary. 
VOL. XXII 2/8/99 2 B 
