258 CELESTIAL MECHANICS. I. VU. 27. 



to the centre of gravity, will be x' — X, y' — Y, and z' — Z; 

 consequently the rotatory inertia with respect to an axis 

 parallel to z\ and passing through the centre of gravity, 



will be S ^{x'~Xf-\-{y'—Yf\.Bm', now, by the pro- 

 perties of the centre of gravity we have S/DwzzmX, and 

 S y'nm=mY; consequently S{x'^-^2x'X + X^+y'^—2yY 

 — y"2).Dm=:— m(X2+F2) + S(y2^/2).D^|-^ Xand F be- 

 ing invariable in the integration]. We may thus obtain 

 the rotatory inertia of the solid with regard to an axis pas- 

 sing through any point whatever, when it is known with 

 regard to the axes passing through the centre of gravity : 

 and it is obvious that [when X and Y vanish, and the 

 centres of motion and of gravity coincide, the rotatory 

 inertia with respect to the centre of motion is only equal 

 to that which belongs to the centre of gravity, exceeding it 

 in other cases by m {X^+Y^), so that] the minimum of the 

 rotatory inertia takes place with respect to one of the prin- 

 cipal axes passing through the centre of gravity. 



351. Theorem. If the rotatory inertia 

 with respect to two of the principal axes of a 

 soHd is of equal magnitude, it will also be the 

 same for any other axis situated in the same 

 plane with them. 



1( AzzB, we have CzzJ sin ^9 sin V + JB sin H cos «(p 

 -\-C cos ^(pzzA sin ^9-{-C cos^; and whenever the new 

 axis zf is in the plane of x" and y'\ it forms a right angle 

 withz'', and C'=i:.4. 



It is easy to understand that in this case, for the axis z" 

 and for any two j/, and y', that are perpendicular to it, we 

 have S x'l/DmzzO, SarVomizO, and S yV'Dm=i:0, for 



