30 Proceedings of Royal Society of Edinburgh. [dec. 20, 
considered is c. BAB' is a semicircle with centre O, and the other 
parts of the curve are symmetrical with respect to OA. If 0 A = a y 
the polar equation of BD, referred to OA as prime radius, is 
supposed to he 
A tt r . 2a 2 .-.. 2 a*... 
6 ~ 2 a Sm 6 + C23 ,^ sm 26 " 3X5 ^ Sm 4 6 
+ 6jjr6 si ' l6e ~ &C,=0 ' 0) 
Then the resulting irrotational motion of the liquid, supposed to 
he in two dimensions only, can be completely determined. It may 
he remarked that the series in (1), as well as all the like series in 
this paper, can he presented in a finite form, hut, as the results are 
rather complicated, perhaps it will he better to leave them as they 
are. It does not matter whether we suppose the solid moving with 
velocity c through the liquid at rest at infinity, or whether we 
suppose the solid at rest, and the liquid moving past it with a 
velocity which at infinity is - c. For simplicity, we shall suppose 
the latter case. First, considering the reflection of the liquid 
motion only from the semicircle BAB', let u and v be the velocities 
of the fluid at any point P(a?, y) or (r, 0) of the fluid in the plane 
of the paper. Let us assume 
U = 2 ^2 cos mO - c , V = 2^5 sin » 
where a m , &c., are arbitrary constants. It is well known that these 
expressions satisfy the hydrodynamical equations. 
Then the velocity in the directions OP is u cos 6 + v sin 0, and 
Cl 
u cos 6 + v sin # = 2 ^ cos (m - 1)0 - c cos 0 . 
B r ow expand cos 6 by Fourier’s theorem in a series of cosines of 
even multiples of 0 between the limits 0 and \tz of 0, which expan- 
sion we observe will be true even at both limits. Then 
a 2c 4c n ~ w 
u cos 6 + v sin 0 = cos (m - l) 6 1 % cos 2 nO . 
* X 7 7T 7T n=1 
cos mr 
4 n 2 - 1 
Now determine a m , &c., by the condition that u cos 0 + v sin 0 
