521 



"These things are evident from considering the trace of the con- 

 stant line OI on the surface of the ellipsoid, and observing that, in 

 general, its projection on the plane of the greatest and least axes is a 

 hyperbola, and its projections on the other two planes ellipses. 



" 2. But if 01 be equal to the mean semiaxis6(aand c being the 

 greatest and least) it will always intersect the body in the same plane, 

 and the elhpsoid in a circle. For there are two circular sections 

 through the mean axis; and therefore if the point I be at any instant 

 in either of them, it will remain in it during the motion. It would 

 be easy to show also, that the axis of rotation, connected with OI by. 

 the construction in the proposition, will in this case always remain in 

 a given plane within the body. 



" 3. If two axes (or moments) of the ellipsoid are equal, OI will 

 describe in the body a conical surface round the third, and the 

 axis of rotation will always be in the same plane with 01 and that 

 third axis, and these three lines will make constant angles with each 

 other ; also the perpendicular on the tangent plane, and therefore the 

 angular velocity, will be constant. These things are evident merely 

 from considering that the ellipsoid becomes one of revolution. 



" 4. Whatever be the forces applied to the body, the varying plane 

 of the maximum of areas and the axis of rotation are always con- 

 nected by the construction in the proposition. But the angular 

 velocity is no longer inversely as the perpendicular. 



" To find the axis about which a body restrained by a fixed point 

 O, and acted on by given forces, will begin to revolve, is usually 

 considered a problem of great complexity. But it may be elegantly 

 solved by means of the ellipsoid described above. Reduce the given 

 forces to a single one through the fixed point O, and a pair ; raise a 

 perpendicular to the plane of the pair from O to meet the ellipsoid 

 in I ; a perpendicular from O to the tangent plane at I will be the 

 initial axis of rotation. 



" The construction is true whether the forces that set the body 

 in motion be impulses or pressures. If they be impulses, and no 

 external forces subsequently act on the body, the axis of rotation 

 will vary its position both in the body and in space ; its course in the 

 body is determined by the preceding theorem, as well as the variation 

 of the angular velocity. The motion of the body in space depends 

 mainly on the two following theorems and the rectification of the ellipse. 



