. 523 



Hence the locus of P is an ellipse whose semiaxes a' and b' are 



and therefore its area 



,^, _ ^bc (a^ - k^) 



-\/(a2_ft2)(a2_c2) 



" But 1 ought to have mentioned, as part of Theorem II., the me- 

 thod of determining in general the areolar velocity of P. Let the 

 angular velocity multiplied by the cosines of the angles which the 

 axis of rotation makes with OA, OB, OC be denoted (as is usual) by 

 />, o, r. These have constant ratios to the perpendiculars, as IP 

 drawn in that proposition ; and in that case the areolar velocity of P is 

 equal to f (6^ — k^^q : and similarly when AOC is any of the other 

 principal planes. Hence, in the present instance, the areolar velocity 

 — i (ct^ _ k?-)p ; and therefore the time of an oscillation 



25r6c 



from the above value of the area of the ellipse, and observing that, 

 since OI is indefinitely near OA, IP and therefore p may be re- 

 garded as constant. 



« If T denote the time of one revolution of the body round its 



axis, then ultimately T = — , and therefore the time of an oscilla- 



' P 



tionis to the time of a revolution as the rectangle under the semiaxes 

 of the section BOC is to the rectangle under the eccentricities of 

 the other two sections, A similar theorem holds when the body re- 

 volves round a line indefinitely near the least principal axis. The 

 times of small oscillations of different magnitudes are equal, as in the 

 pendulum. 



" Many particular consequences might be deduced from what has 

 been said ; but it will be better to mention some new theorems 

 about moments of inertia and centrifugal forces. 



