288 Mr. Booth's Remat-h on a Statement by Poisson. 



A, B, C, while the moments of inertia round the principal 

 axes O x\ Oy, O Z are 

 A', B'j C ; through the centre 

 of gravity and the point O, 

 let any two arbitrary parallel 

 right lines be drawn, G m and 

 O n ; let the moments of in- 

 ertia round these right lines 

 be D and D', and put the 

 distance GO = c. 



Let the angle ZGm or 

 ZOrc be 0, the angle be- y 

 tween the planes Z G X and " 

 Z G m — <J>, then we shall have 



D = A sin 2 cos 2 <p + B sin 2 sin 2 <f> + C cos 2 (I.) 

 But as the line O n is parallel to G m and distant from it by 

 the distance c sin 0, we shall have (Traite de Mecanique, 

 torn. ii. page 53.), M being the mass of the body, 



D' = D + Mc 2 sin 2 0, or putting for D its value, 



D' = A sin 2 cos 2 <J> + B sin 2 sin 2 <J> + C cos 2 \ , TT v 



+ Mc 2 sin 2 Jl* ! 



The angle between the planes Z O n and ZO^ is (<$> — co) : 

 referring then the moment of inertia D' to the moments of in- 

 ertia of the three principal axes passing through the point O, 

 we find 



D' = A' sin 2 cos 2 (<f> — co) + B' sin 2 sin 2 (<f> — co)" 

 + Ccos 2 . . 



Equating the values of D' given by equations (II.) (III.), 

 eliminating and dividing by sin 2 0, there results 



A cos 2 <J> + B sin 2 $ + M c 2 = A' cos 2 ($ - co) \, TV v 

 + B' sin 2 ($-») Jl lv -J 



We have now to determine A', B'. 



Through G let a right line G t be drawn parallel to Ox'; 

 the moment of inertia round this line is A cos 2 co + B sin 2 co, 

 and as this line passing through the centre of gravity is distant 

 from Oi'' by c, the moment of inertia round O/ or A' is 



equal to 



A cos 2 co + Bsin 2 co -J- Mc 2 . Similarly, 



B' = A sin 2 co + B cos 2 co + Mc 2 . 

 Substituting these values of A' and B' in (IV.), we find 

 A cos 2 4> + B sin 2 <f> = A {cos 2 co cos 2 (cf> -co) -f sin 2 col 



sin 2 (<$>-«>)} + B{sin 2 «cos 2 (<f>— co) + cos 2 co I (V.) 

 sin 2 (4>-»)} J 



} }(ni.) 



