1076 Proceedings of Royal Society of Edinburgh. [sess. 
§ 84. For interpreting and approximately evaluating the definite 
integral, we may conveniently put 
M jWTf, and tt = 008 (’/' - *> . . . (112), 
V y ’ COS 2 ^ 
and write (111) as follows : 
**v)-*#> P x2^ sin2 F • - (113) - 
Now if we suppose r/X very great, there will be exceedingly 
rapid transitions between equal positive and negative values of 
sin (fhrrujX), which will cause cancelling of all portions of the 
integral except those , if any there are, for which du/di Jr vanishes. 
We shall see presently that there are two such values, i/q, x J/ 2 , both 
real if tan0< Fil u being a maximum (uf) for one of them, 
and a minimum ( u 2 ) for the other ; and that, when 0 has any 
value between tan -1 and - tan -1 the values of xj/ v x J/ 2 
are both imaginary. Consideration of this last-mentioned case 
shows that, in the whole area of sea in advance of two lines 
through the centre of the travelling forcive inclined at equal 
angles of tan -1 (or 19° 28') on each side of the mid- wake, there 
is no perceptible disturbance at distances of much more than a 
half wave-length from the centre of the forcive. The main 
disturbance by ship-waves, therefore, lies in the rearward angular 
space between these two lines. It is illustrated by fig. 32, as we 
now proceed to prove by the proper interpretation of (113). 
Expanding the argument of the sin in (113) by Taylor’s theorem 
for values of xj/ differing from xf/ 1 by small fractions of a radian, 
we find 
27 rru , 
. 27rrr 
aT = 
” A L 
^xh + <$), ■ <i»> 
where 
df 
From the second of (115) we find dxf/ = dq 1 /(^ 1 Jtt), where 
dxf/ 2 Ji 
(116). 
Dealing similarly in respect to x[/ 2 and values of xf/ differing but 
