. ITY. 



. w&V sin o> sin a. I x { 1 - -a] <fcc 



. An' \ O, I 



x = 



I 

 n a> sin a 



Now (1) gives a constant value for X, and then (2) shews 

 that x bears a constant ratio t< 



Thus the locus of the centre of gravity of segments of an 

 ellipsoid of constant volume is an ellipsoid similar to the. 

 original ellipsoid and similarly situated. 



(7) Find the centre of gravity of a portion of an ollij 

 comprised between two cones whose common vertex is at the 

 centre of the ellipsoid and whose bases are parallel. 



The volume between the two cones may be divided into an 

 indefinitely large number of shells which have tin- cent 

 the ellipsoid as their common vertex, and their basefl in planes 

 parallel to the bases of the two cones. AVc shall first shew 

 that if the planes which contain the bases of the shell 

 equidistant tlie shells are all equal. Take conjugate semi- 

 diameters as axes, and let the plane of (y, z) be parallel to 

 the bases of the two cones. The volume of the cone which 

 has the centre of the ellipsoid as vertex, and for its base the 

 plane curve formed by the intersection of the ellipsoid with 

 the plane which has x for its abscissa, is 



sn o> sn o 



/ x*\ 

 f 1 -- r 2 J x, 



where the notation is the same as in the preceding example. 

 The volume of the cone which has the centre of the ellipsoid 

 as vertex, and for its base the plane curve formed by the 

 intersection of the ellipsoid with the plane which has x+ A# 

 for its abscissa, is 



sn 6> Sn a 



jl - kMj ( x + Ax). 



