32 Proceedings of the Royal Society of Edinburgh. [Sess. 
Substitution in (4') gives 
4cy(c 2 + 1 ) -n --u M+o>t) 
* 4yc+(c 2 -l) 2 
_2(c 2 -l)(c 2 + l) 
V ~ ±yc + (c*-iy ^ 
The component displacements in the original ray are, for x = 0 , 
(S') 
L = B ibe ib{y+at) 
Vo 
- B 
and hence 
l = + 
2L= + 
Vo 
4cy(c 2 + l) 
4yc + (c 2 - l) 2 
2(c 2 - l)( c 2 + 1) 
4yc + (c 2 - l) 2 
}• 
r 
So long as c 2 does not become less than m/w, these ratios can be 
calculated as in the former case ; but when c 2 is less than m/n, y becomes 
imaginary, and the expressions for £ and rj must be modified. For this 
purpose we write y~ie) and, putting for convenience b(y 4- cot) = 0, we 
find for the component displacements the values 
j= _ 4ce(c 2 + 1 ) B&j 0 
C 4«€C + (c 2 - l) 2 
2(c 2 — l)(c 2 4- 1) -p, . ie 
V= ~ 1 V c + g 2 -!) 2 ^ 
( 6 ) 
These become 
4ce(c 2 +1) 
1 6e 2 C 2 4- (C^iy 
2(c 2 - l)(c 2 + 1) 
v ' i6A 2 + (c 2 - l) 4 
the real parts of which are 
m - 
4ec(c 2 + 1) 
16e 2 c 2 + (c 2 - l) 4 
B5((c 2 - l) 2 - 4^€c)(cos 0 + i sin 0) 
B b((c 2 - l) 2 - iuc)(i cos 6 - sin 0) 
{(c 2 - l) 2 cos 0 4- 4ec sin #}B6 
2(C 2 - 1)(C 2 + 1) r . a ivn • 
V = - o 9 ,/ 9 — R4ec cos 0 - (c 2 - l ) 2 sin 6$Bb 
If we put tan p — 
16e 2 c 2 + (c 2 - 1 ) 
4ec 
- we get the simpler forms 
1 
0 2 -i) 2 
£= - B b(c 2 + 1 ) sinjp cos ( 0 -p) 
r] = + 2B5 ^ c 2 + j ^ cos p sin (6 -p) (" 
c — 1 J 
• (0 
