275 T. H. Havelock. 
together with a trail of doublets to the rear of C,. The image of this system in 
the circle gives a doublet — Ma?/4f? at C,, together with a certain line distribu- 
tion of doublets on the semicircle on CC,. 
So the process could be carried on, but we 
shall stop at this stage. 
From the results already given, we could 
build up complete expressions for the velocity 
potential and surface elevation for each 
stage. It would be of interest to work 
these out graphically to compare with fig. 1; 
but the expressions soon become complicated 
and their evaluation difficult, especially for 
the immediate vicinity of the origin. We 
shall therefore limit the study to the regular 
waves established in the rear of the cylinder. We have seen that the regular 
waves of the first approximation, due to the doublet ca? at C, are given by 
y = 4rxpae~"" sin Koz; Ss 70. (18) 
We take the next stage in two parts. First we have an isolated horizontal 
doublet of moment — ca*/4f? at C., whose co-ordinates are (0, —f-+ a?/2f). 
From (11) it follows that the contribution of this doublet to the regular waves 
is 
Q = —TKpa'f 2e OF -P) sin noo; 2<0: (19) 
Next we consider the line distribution of doublets to the rear of ©, and its 
image in the circle. Referring to the results in § 2, there is at the point (—p, f) 
an elementary doublet of moment 2x ca*dp, with its axis making an angle 
kop — 4m with the positive direction of Ox. The image of this in the circle is a 
doublet at the point whose co-ordinates are 
a*p earn 
= era ali ; 20) 
pep lpr P 
the moment of the doublet is 2« ca* . dp/(p? + 4f?), and its axis makes with 
Oz the angle 
2 tan™ (p/2f) — kop + 3m. (21) 
From (11) we can now write down the waves due to this doublet. It should 
be noted that the expression will hold for 
LEDS 
e+ eae <0. 
