28 
Proceedings of the Royal Society of Edinburgh. [Sess. 
spectively with the wave-length and the period ; i is the ordinary 
imaginary of analysis. 
The equations of motion (I.) in the two media give 
(m + n)(c 2 + l)= p0 ) 2 = n(y 2 + 1) \ , 
k'(c 2 + 1) = m(c 2 + 1 ) •= p'o) 2 j 
The conditions to be satisfied at the surface are : — 
(1) Equality of normal displacement on each side of the interface, 
i=£> ° r 
a i + ^ = M when x = 0 . 
dx dy dx 
(2) Equality of normal stress on each side of the interface, P = P', or 
(m + n) V 2 of> - = \/ 2 cf>' when^ = 0. 
\dy 2 dxdy) 
(3) Equality of tangential stresses on each side of the interface, 
U = 0, or 
2 aV a^_sV =0 when;c= o. 
dxdy dy 2 dx 2 
These lead to the equations 
B x + c(A- A x ) 
c'A' 
-2 y B 1 + (/-l)(A + A 1 )-£( 7 * + l)A' \ 
(y 2 - 1 )Bj - 2c(A - A,) = 0 
(3) 
The object of the inquiry is to find the normal and tangential displace- 
ments at the surface, i.e. the values of £ and y when x = 0. These values 
are, when x = 0, 
( = d 4- + — = {c(A - A ,) + B 1 }«V i W“ , > 
dx dy 1 v 17 11 
V = ^ ^ = { A + A, + yB, } ibe ib <»+‘“> 
dy ox 
(4) 
These give at once 
£_ c(A - A x ) + B 1 
y A + S + yBi 
1 y 2 - 1 
2y + t}(y + 1 ) 
p c \ yj 
by substitution from (3). 
For the case of rock and air we may put 
2000 p' = p , 3n = m+n = 9 x 10 11 , Af = l’5xl0« 
