1887.] Sir W. Thomson on Procession of Waves . 
43 
and by (22), (3), and (2) we find 
1 [ (r + y + b) 1 . 
9 
, , sin ait } . . . . (25). 
dt r j v 7 
What we chiefly want to know is the surface-value of rj , which 
we have denoted hy - f) ; and we shall work this out for the case 
5 = 0. But it is to he remarked that the assumption of 6 = 0 does 
not diminish the generality of our problem, because the motion at 
at any depth, e, below the upper surface with 6 = 0, is the same as 
the motion at the surface, with b = c. 
Put now 6 = 0 and y = 0 in (15) : we find 
gt 2 . gt 2N 
= * *( C08 E + sin e) - sin ('E + % 
Using this in (24), and putting a 2 — gf 2 /ix, we find 
(26). 
= -2 J- f 
V gjo 
■2 nVL 
gj o. 
da- 1 COS 
cos 
dcr sin g wt - 2 
Vr) si 
Sill I o~ + 
7T 
(27), 
[("VD"- 
d ?\2 x t r 
— or h co£ + -r 
S' 4 - 
7 r 
or- co£ + = 
.</ 4 _ 
(28). 
Using now the following notation. 
say 6 = j dO sin 0 2 j cay 6= f dO cos 0 2 . (29), 
J 0 J 0 
for two integrals which have been tabulated by Airy * through 
/ (£) 
the range from 0 to 5*5^ ^ we reduce (28) to 
+ 
cos( — — <o£ — — 
\* 4 . 
. / (id TV 
Sin g — X — C)t— T 
\<7 4 . 
lor , (a 
cost X + 0 it - — 
\g 4 , 
or (ii 1 
— a? 4- (at — 7- 
^ 4 > 
Tracts” (Undulatory Theory of Optics, last page). 
