10. Investigate an equation for the form of the floats in the paddle
wheels of a steam vessel in order that they may enter the water without
splashing.
218 EXAMINATION PAPERS FOR [Jan. 19,

If u s hw y where u = the velocity of the vessel, w = the angular velocity
of the wheels, and h = the height of the centres of the wheels above the
water, shew that the floats of each wheel must have the forms of arcs
of involutes of a concentric circle touching the water level.
11. A hollow vertical polygonal prism, open at both ends, rests upon
a horizontal plane. Every two contiguous faces are moveable about their
common edge. Supposing the prism to be in equilibrium, when filled
with fluid, prove that

c i _ g t = c s _.
sin a, sina, sina» ""'

a v a„ a„ ... being the angles of a transverse section A l A a A s ...A n A v and
c X t c v Cp ... denoting the lines A„A V <A X A V A % A ...
Hence shew that there will be equilibrium when the points A v A v A#...
lie all in the circumference of a circle.
12. A filament of fluid oscillates in a thin cycloids! tube of uniform
bore the axis of the cycloid being vertical and its vertex downwards.
Supposing the filament to be placed initially with its lower end at the
lowest point of the tube, find the pressure at any point of the filament at
any time.
Shew that the pressure is a maximum, during the whole motion, at the
middle point of the filament.
