Images in Some Problems of Surface Waves. 272 
are due to the semi-infinite train of doublets behind the image point. Part of 
the effect is the train of regular waves to the rear of the origin, evidently associ- 
ated with the periodicity in the direction of the doublets along the line 
distribution ; and there is also a further contribution to the local surface dis- 
turbance, which we may regard as arising from the fact that the line distribution 
is semi-infinite and has a definite front. 
Horizontal and Vertical Doublets. 
3. With the axis of the doublet horizontal, we have the well-known first 
approximation to the submerged circular cylinder of radius a, if we take 
M = ca?. From (11), the surface elevation can be expressed in the form 
2 
— 20, + 2a?%xoP, «> 0, 
y= ee + f? 
= _2a'f + 2a7KoP + Arxoa2e—! sin nox, 2 <0 (12) 
a + f? ? 2 
where P is the real part, for 2 > 0, of the integral 
20 e tim 
| ae EE (13) 
0 % + UKo 
Taking the axis of the doublet to be vertically upwards, we have « = x/2 
in the general formule ; and, putting M = ca? in this case also, we obtain 
2a*x ° 
7= cee +f = 207K Q, z> 0, 
2a°a 2 2,—Kof 
i eae + ft + 2a%KoQ + 4rKoa%e—"! cos xox, x <0, (14) 
where Q is the imaginary part of the integral (13). This integral may be 
expressed formally in terms of [i (e’—*), where li denotes the logarithmic 
integral, and may be expanded in various forms. For the numerical calcula- 
tions which follow, it was found simplest to use the series 
> o— (atip)u 
| 0 u+t 
A=y-+logr +E cos nO, 
1 4 
du = — (A + aB) e=#-i®, 
Boe) dn ab (15) 
in! 
where 
r= («? + B?)}, tanO=a/B, and y=0-57721. 
269 
