Bhatt T. H. Havelock 408 
variable and integrating round suitable contours according as x is positive 
or negative. We obtain, taking the real part and making p zero, 
= COS of -F 2k y7Me™ “ of" )cos (otk a) 
= ue cos of ++ 
ist 2 
Ky Cos K(f—y)-+K sin «(f—y) eF de, (3) 
K+ Ko? 
the upper or lower signs to be taken according as 2 is positive or negative. 
The corresponding surface elevation 7 is given by 
2k M cos at | 
0 
2 t 
n= aL ein (ot Fk yx)— aie cin at 
g Gg Bay 
4 Peco i | eocosil i Bin of eF die. (4) 
g 0 K"-+-Ko 
The first term represents the regular waves, while the other two terms 
give a local oscillation whose magnitude diminishes with increasing 
distance from the centre of disturbance. 
Similar expressions may be obtained for a source of oscillating mag- 
nitude, or for a doublet with its axis in any direction. It may be remarked 
that for a doublet at a given point in the liquid, so far as the regular 
waves are concerned the direction of the axis affects only the phase of 
the waves and not their amplitude. 
3. Consider the motion produced by an elliptic cylinder moving through 
an infinite liquid. If the motion of the cylinder is one of translation, 
it is well known that the fluid motion is that due to a certain distribution 
of doublets along the line joining the foci of the elliptic section of the 
cylinder; a similar proposition may also be readily proved when the 
motion is one of rotation. 
In particular, let the cylinder be moving with velocity V parallel 
to the minor axis of the section ; let S, S’ be the foci of the section and 
h the distance of a point on SS’ from the centre C. The moment per 
unit length of the doublet distribution is 
MPS Vid@=O), 3 5% 6 = « 6 @&) 
in the usual notation, the axes of the doublets being perpendicular to SS’. 
If the cylinder is rotating round C with angular velocity w, the moment 
per unit length along SS’ is 
w(a+b)h(a%e2—h2)t/2n(a—b), . . . . . . (6) 
the axes being perpendicular to SS’. 
Combining (5) and (6) with a suitable value of V, we may obtain the 
distribution when the cylinder is rotating about any point on the major 
axis, 
463 
