+, &c. (fi 



192 



inequalities which produced it were more considerable; only it will 

 take more time to arrive at a permanent state. 



To calculate, then, the effect of the disturbing force, we must de- 

 velop the right-hand members of equations (A). To begin with the 

 first. The quantity to be developed may be put in the form 



2m (zy x - yz x ) { (x 2 + y 2 + z 2 ) - 2 {xx } + yy x + zz x ) + xf + y* + z x 2 j " 4 

 or _ 2 m (zy x - yz x ) [l-<2 + — 



== -^ m (zy 1 -yz 1 )[l+3 + 



15 (xx x + yy x + gg x ) 



2 r 4 



1 3 

 = —2m (zy, - + (zy, - yz,) (xx x + yy x + zz x ) 



1 3 



= — (z2 (my,) - y2 + — [zx 2 (ma;,^) 



+ zy 2m (y, 8 - z, 2 ) + (z 2 - y 2 ) 2m ( y x z x - #y 2(m z x x x ) 



rejecting for the present the further terms in the development. 



By the property of the centre of gravity and the principal axes, the 

 terms 2(m y,), &c, and also 2(m x x y x ), &c, vanish ; and it is re- 

 duced to 



3 3 



— zy S.m {y? - tt'), or — zy 2m (y* + xf - (s, 2 + x, 2 )) 



Izy(B-C), 



hence the first of equations (A) becomes 



div, B-B 3/i C-B 



similarly div 2 A - C Bp A - C 



— h — 00,00, = ZX 



dt B 3 1 r 5 B 

 dio 3 B - A Sfi B-A 



and if in these we substitute the values of xy and 2, given above, they 

 will become 



rfWl C-B 3^ C-B. /1 + cos* 



dt + — ^ = 2 73 72~ Sm ' p2~ C ° S (0 " ^ 



, 1 - COS I , 'J: 



cos 1 cos 0 — cos {<p + 29) 



