1886.] Rev. H. J. Sharpe on Motion of Compound Bodies. 33 
Assume for a solution of this equation 
dd> — mp . 
f— = 2iCi m e sm ma , 
da 
d(f) ^ -mi 3 
- irt = 2iCi m e cos ma -l- G , 
dp 
where C is a constant, and a m , &c. are constants to be determined. 
We may observe that the velocity at infinity normal to the ellipse 
/3 = const, is dxfijVdp where P 2 = |c 2 (cosh2/3 - cos 2a), so that whether 
C is zero or finite, with the above assumption, the whole velocity at 
infinity is zero. Now suppose /3 = /3 1 gives the solid cylinder, then 
it will be found that the condition for reflection of the fluid motion 
at /3 = /3 1 is 
^a m e "^cos ma + C = Yc sinh J3 1 cos a (3) 
If the ellipse is complete, (3) must hold for all values of a from 
0 to 7 r, and then we shall find that we must have C = 0 and m = 0. 
This leads to the solution given in Art. 88 ( d ) of Lamb’s Motion of 
Fluids. But if the reflection takes place at only a quadrant of the 
ellipse, (3) admits of another solution. Expand in (3) cos a by 
Fourier’s Theorem in a series of cosines of even multiples of a from 
a = 0 to a = Jtt, and we get 
2 cos 7 m 
Now let 
^ 2 f od z cos 7 m ) 
cosma4-G = - Vcsmh/jj < 1 - cos 2^a _ \ ( • 
m = 2 n, 
and 
C = — Vcsinh R, 
7T ' L 
a 2n e 2nl3l = - — Ycsinh/3 
cos mr 
'Hn 2 - 1 ’ 
and we have a solution exactly analogous to the case of the circle. 
It will be found also that instead of taking a quadrant of an ellipse, 
we might take AB to be any arc and still get a similar proposition. 
We will next take a case of motion in three dimensions. We 
will suppose a solid hemisphere BAB' of radius a moving through 
an infinite mass of liquid in direction OA, with a velocity which at 
the instant considered is V, the velocity of the liquid at infinity 
being zero. The motion of the liquid is supposed to be in planes 
through OA, and to be symmetrical round it. We will suppose a 
VOL. XIV. 20/8/87 c 
