1887 .] Sir W. Thomson on Procession of Waves. 
45 
This represents a uniform procession of free waves, of which the 
wave-length, A, and the wave-velocity, U, are as follows : — 
A = 27ry/a) 2 , U = ^/co (40). 
To explain the meaning of “ very large ” as we have just now 
used it, let 
x 
= n\, which makes = ij^n, and— l/4?r J n (41) . 
Hence the term of (38) omitted in (39) is 1/47T Jn of that retained. 
And the value of the R, omitted by (36) in (37), is of the order 
1/2 J2 n of the Q which is retained, because 
cay(co ) — cay( J2i m)= 
sin (27m) 
2 
and say{o o ) — say( J2 7m= 
cos (27 m) 
(42), 
2 J 27m 
when n is very large. 
In (36) and its consequence (31), we supposed if so large 
that M is large positive ; let us next suppose t so 
small that it is large negative; that is to say, let 
t = 2o>x/g-m ^ — (43), 
where m is a large positive numeric. Thus, remarking that cay 
( - 6 )= - cay ((9), and say ( - 0) = - say ( 0 ), we have, by (43) and 
(41) in (32), 
Q= P^{[ ca vi m ) 
if 
- cay( rn)f + [say (m) - say( J27ni)ff (44); 
and therefore, when m and n are each very large, Q=0. Because 
n is large we still, as in (36), have R=0; and therefore the motion 
is approximately zero, at any considerable number, n , of wave- 
lengths from the origin, so long as m in (43) remains large. As 
time advances, m decreases to 0, and on to — oo : and, watching 
at the plac ex = n\, we see wave-motion gradually increasing from 
nothing, till it becomes the regular procession of waves represented 
by (39); and continues so unchanged for ever after. When m = 0, 
that is to say, at the time 
t = 2c or/g . 
• • (45), 
