593] 
241 
593. 
A SHEEPSHANKS’ PROBLEM (1866). 
[From the Messenger of Mathematics, vol. iv. (1875), pp. 34—36.] 
Apply the formula) of elliptic motion to determine the motion of a body let fall 
from the top of a tower at the equator. 
The earth is regarded as rotating with the angular velocity &> round a fixed axis, 
so that the body is in fact projected from the apocentre with an angular velocity = w; 
and we write a for the equatorial radius, ¡3 for the height of the tower; then g 
denoting the force of gravity, and g, h, n, a, e, 6, as in the theory of elliptic motion, 
we have 
g — n*a 3 = go?, 
h = (a+ a) = na 2 V(1 — e 2 ), 
ol + /3 = a (1 + e); 
whence 
(a + /3) 4 to 2 = go?a (1 — e 2 ), 
(a + /3) = a (1 + e ), 
(a + ¡3 f ft> 2 = coo- B 
gc? g \ a. 
where — = ratio of centrifugal force to gravity, 
9 
> 
so that 1 — e is small; 
q(l — e 2 ) _ (a+ B)(l —e) 
1 1 — e cos 6 ’ 1 — e cos 0 
whence 
1 - e cos 6 
(a + /3) (1 - e) 
r 
C. IX. 
31