OF THE MOTIONS OF A SOLID BODY. 2.09 



taking x" iind y" for the two given principal axes, we have 

 by the sup^iosition ^{x"^ -V z!'~)^m^^^{i)"~ -^2!"^)ximy whence 

 Sy^D^— Sy«j)/7i: now if b be the angle made by x' and 

 a/', we shall have, as in article 324, x'-=:.x" cos t-\-y" sin ?, 

 y':=^y" cos £ ~x" sin f, whence Sx'y'Diii =z Sx"y"Dm (cos-e 

 — sin*f) + S(y"2 — j:'"2)Dwi sin cos eznO. And in the same 

 manner it may be shown thatjSaV'DmrzO, 2indSy'z"DmziO; 

 so that all the axes perpendicular to 3" will be principal 

 axes; and their number will be unlimited. 



352, Corollary. If A=:B=C, we have 

 in general C=A, and the rotatory inertia is 

 equal fof every axis. 



We have here S x'yDm—0, SarVD/wizO, S yVDmnO 

 whatever may be the position of the axes x' and y\ so that 

 all the lines passing through the centre of gravity are 

 principal axes. This is the case with the sphere, and we 

 shall hereafter find that the property belongs to an infinite 

 number of other solids, of which the general equation will 

 be demonstrated. 



§ 28. Investigation of the momentary axis of rotation of 

 a body : the quantities, which determine its position with 

 respect to the principal axes, give at the same time the 

 velocity of rotation. P. 79. 



353. Theorem. There is always one axis 

 at rest, in every body of which any point is at 

 rest, although the same axis may only be at 

 rest for a moment. 



[We may readily conceive the nature of a momentary 

 axis, by considering that a rolling cylinder revolves round 



s 2 



