460 Mr. H. W. Chapman on 



Also by the modified form of Euler's equations, 



A —~ — A 3 co 2 4- C6 2 co s = moment of external forces about GA, 

 A~j^—Cj6 1 uj z -{-A0 z (d 1 — „ „ „ GB, 



1!^-A^ + Afe= „ -, „ GO'. 



These give by (1) 



—A^s'm&-2A0yjrcos0 + Cco 3 = F 2 (h + acos0), . . (2) 

 A0 + Cft) 3 "^ sin 0- Ayjr 2 sin cos = R7i sin — F 1 (a + h cos 0), 



■ • (3) 

 Ca>3 = F 2 asin<9. . . . (4) 



Next consider the accelerations of G referred to OX, OY, 

 OZ. This system of axes has a spin yfr round OZ, so we get 



du . F, ... 



^ -^=M' W 



^ • F 2 



rf< + **+•= 5P w 



cfoy R 



a- = m^ ( 7 ) 



We have also the geometrical equations arising from the 

 fact that has no velocity, since the surface is supposed to 

 be perfectly rough. 



Velocity of along OX = velocity of G parallel to OX 

 -f velocity of relative to G 



parallel to OX 

 = u — w 2 (a + h cos 0) , 

 .-. from (1) u = (a-\-h cos 0)0 (8) 



Velocity of along OY = velocity of G parallel to OY 



+ velocity of relative to G 



parallel to OY 

 = v -f co^k -f a cos 0) -i- co B a sin 0, 

 .'. from (1) v = -*|rsin0(/i + acos#) — co 3 asm0 (9) 

 and clearly w = —h sin 00 (10) 



4. As the plane is perfectly rough we can use the equation 

 of energy. This is 



iM(u 2 + v 2 + w*) + i ( A©! 2 + Aco 2 2 4- C&> 3 2 ) + Mgh cos = constant. 



