.REPRESENTATION OF THE MOTION OF A FREE RIGID BODY. 
V 
each of these forces acts at an arm ^ Hence 
x 
P 
dt 
! 2p 2 dp. . 
= G Tf‘s b y(0: 
F=2G i; |. 
(«) 
2\ 
It is worthy of notice that F= — h t . 
Again, 
r,P dht 7 
v W=Tt ~ nll < 
r f l 9 \. 
= G 'iXlF- n P‘) ’ 
••• p = g f(^-4 • 
(10) 
It may be desirable to replace these expressions by others in which p, shall be the 
only variable quantity, and which shall be free from differential coefficients. This 
may be done as follows. Writing, for shortness, «, (3, y in place of 
.. a 2 IP n c 2 
1 o 5 1 o 5 1 5 
p p 
respectively, it may be proved, from equations (1), (2), (3), (4), that 
2 
Pit) =- O 2 + (ly^)O 2 + y^ 2 ){p 2 + «/3a 2 ) (11) 
Again, it is proved by Poinsot, ‘ Sur la Potation des Corps,’ p. 130 (see also Quarterly 
Journal of Pure and Applied Mathematics, vol. vii. p. 74), that 
p=X + ci(3 y-g, 
r* 
( 12 ) 
a result which also follows from (5), (6), (7), remembering that the angular momentum 
o 
of the particle about the normal to the rough plane is p?n. 
Now, differentiating (11), 
j t (^'ITt) = — (^ 2 +y«A 2 )(yJ+«/3A 2 )— (^ 2 +«(lA 2 )(^ 2 +^y^) 
— (T 2 +,3y/v)(yJ + yal:) ; 
(c 2 + y«* s )(^ a + «/^ 2 ) — <> 2 + «/^ 2 ) (A + P*A 2 ) — (A + /V- 2 ) (A + y°^ 2 ) 
