OF THE MOTIONS OF A SOLID BODY. ^1^1 



vious that the angle fl is 

 equal to the angle formed 

 by the two axes perpendicu- 

 lar to the planes of which it 

 is the inclination; and the 

 projection of 2! on the equa- 

 torial plane of the body, or 

 on that of x" and y\ being z' 

 sin 5, and the perpendiculars falling from its extremity, on 

 x" and on y", z sin 5 sin (p and 2' sin 5 cos <p respectively, 

 these perpendiculars will be the cosines of the angles 

 formed with 2', when 2' is the radius. ] 



Scholium 2. Hence it follows in general, that if we 

 multiply the rotatory inertia belonging to each principal 

 axis by the square of the cosine of the angle which it 

 makes with any other axis, the sum of the three pro- 

 ducts will be the rotatory inertia with respect to this axis. 



349. CoROLLAEY 2. The greatest and 



the least rotatory inertia belong to two of the 



principal axes of rotation. 



For the quantity C is always less than the greatest of 

 the three quantities A, B, and C, [, because their joint co- 

 efficients are always equal to unity] ; it is also greater than 

 the smallest, for a similar reason. 



350. Corollary 3. The minimum of 

 rotatory inertia belongs to one of the principal 

 axes passing through the centre of gravity. 



Let the coordinates of the centre of gravity, reckoned 

 from the common origin at the centre of motion be X, Y", 

 and Z, then the coordinates of the particle Dw? as referred 



