522 



Let the principal axes of the ellipsoid (or 

 a,6, c)be OA, OB, OC. 



" Theorem II. — If a perpendicular IP 

 be let fall from I on any of the principal 

 planes (as AOC), the areolar velocity of P 

 round O will be proportional to the perpendicular IP. 



" By areolar velocity I mean the increment of the area (as P0/>) 

 divided by the increment of the time when taken indefinitely 

 small. 



"Since IP = \/(0P - OP^, theposition of Ol.and therefore of 

 the axis of rotation at any given time, may be determined from this 

 theorem by the method of quadratures ; and it may be reduced 

 to the rectification of the ellipse. 



" Theorem .III. — Let a plane passing through the fixed line OI 

 and any of the principal axes (as OB), intersect the plane AOC in 

 OQ, and the invariable plane (to which 01 is perpendicular) in a 

 straight line which may be called OR ; the angular velocity of OR 

 is inversely as the square of OQ; and hence if OR be always taken 

 equal to OQ, the point R will describe in the invariable plane areas 

 proportional to the times. 



" Since OQ is known at any time by the preceding proposition, 

 the position of OR at any time will be known from this by the me- 

 thod of quadratures. Also the inclination of the plane AOC to the 

 invariable plane is known, since it is equal to the angle OIP. 

 Hence the position of the body at any instant is completely deter- 

 mined. 



" For an apphcation of the theoremslet us take the following prob- 

 lem : — The body revolving round a line indefinitely near the greatest 

 or least of the principal axes, to find the time of an oscillation. By 

 the time of an oscillation I mean that in which 01, and consequently 

 the axis of rotation, returns to the same position within the body. 



" Let the axis of rotation be indefinitely 

 near OA. Then x, y, z being the coordi- 

 nates of I, we have a;^-j-^^ -^- z^ ::z OP ^ A;^, , /CT"^ 



a„d-^-|--^|'-f-^=l. Therefore ^ ^^^ 



