1909-10.] On Waves in a Dispersive Medium. 247 
§ 7. If the whole disturbance, or if the most important part of the disturb- 
ance, f(x), be confined to a small part of the medium on each side of the 
origin, we can simplify further the expression (16) by introducing the condition 
that x is so great that we may consider only the first power of (xjx) for the 
greatest value of x x that need be included in the above integration in obtaining 
a first approximation to the value of f The simplified expression of (16), 
which is true approximately when x and t are both very great, is obtained by 
j L _l n+1 
expanding (x — x x ) 2w 2 and (x — ay) n so far as the first power of x v and then 
transforming the cosine term by the addition theorem. It is : 
£=P 
i__i 
Xln 2 
“3 
P n 
cos 
+ 00 
e (**,/(*,) .( i _ (1 - I Vi ( cos i j£±}(Z.)\ 
| 1 17 ) \2» 2 J X ( ) n \t J 1 
sin ej daij/fe) j 1 - (ttH ij 
n \t 
with 
6 = 1 B— + - 
,! _ 4 
t n 
(17). 
The lower sign is to be taken in 0 when n is positive. 
§ 8. We can now find a second solution of our problem, called for at the 
end of § 4. Introducing the form of initial displacement of equation (1), 
(17) becomes 
+ 00 
y_i 
'y*2/i 2 
fin 
cos 6 dx Y 
-00 
+ 00 
+ sin 0 J dx 1 
-00 
0/1 iN 
1 V2 n 2/ 
It 
cos | 
B n + 1 ( 
' x X 
a 2 + x ^ 
n ' 
dj 
{ -(U 
Wi ( 
)x 1 
_ sin < 
( x X 
a 2 + x j 2 
\ n 
\tj 
x 1 | 
• (18). 
The integrals in this expression belong to the same class as those dealt with 
in § 4, and can all be evaluated, the result obtained when this is done being 
<. -ryTrX^i _w±l/*\n B f A ( 1 l\a ‘ 1 /io\ 
The second term on the right-hand side of (19), being of the order (1 /x) of 
the first term, is negligible compared with it by the condition stated in § 7. 
The displacement at point x due to our initial disturbance of equation (1) 
VOL. xxx. 17 
