Motion of a System of Bodies. 51 



,B, ,C are each=0 : by substituting the values of c", d c from (o) 



A' 



in (35), thev become cos. &=l—j== — — , sin. 6 cos. -X 



B' 



C 



/-r ~ - , sin. 6 sin. -1= , =L=-=r==L, (36). (35) 



agree with the formulae given at p. 269, Vol. 1. Mec. Anal, for the 

 determination of the invariable plane, and (36) are given for the 

 same purpose, at p. 60, Vol. 1. Mec. Cel. and it is evident that they 

 agree with (16). 



Again, suppose the system to be rigid, or that the bodies which 

 compose it are invariably connected with each other; also that x / , 

 y\ z', are invariably connected wiih the system, so that they do not 

 vary with the time, and change their values only in passing from one 

 body of the system to another. 



dx' dy' ■ dz' 



In this case^-=0, -~= 0, ^-=0,.' V A = 0, ,B=0, ,0=0, and 



dp' 



A, B, &c. are each constant, hence (26) will be changed t0 "^7+? r/ j 



a "dN'"+a'dN"+adN' da' b"dN'"+b'dN / +bdN t 



&+ , , c"dN'"+c'dN , '+cdN' ■ x 



f+pff-f- J t , (38). 



It is evident by (p), that the axes of x\y\z' can be found so as to 

 satisfy the equations D = Sy' z'm = 0, E = Sc / z'm=0, F = Sz'y / m 



0; then will the axes of oc/.y',z\ be principal axes. Hence put 

 D=0, E=0, F=0; then by (/'), p' = &p, 9' = B?, r / = Cr-, hence 



Adp a'tdW + a'dW+adN' Bdq ' 

 (38), become ^f+(C-B) ? r= Jt , w + 



b"dW" + 6W + bdN' Cdr en < 

 (A - C)pr = Jt , W + ( B- k)pq 



5 ,(39). 



Since the position of the axis of x in the plane x, y, is arbitrary, 

 we willl now suppose that it makes an infinitely small angle with the 

 line of intersection of the planes x, y, and x', if. hence neglecting 

 infinitely small quantities of the second, &c. orders, we have sin.^ 



4^ cos. 4^=1. '.substituting these values of sin. 4>, cos. 4^ in (0), 

 we have by neglecting quantities of the order 4"> a"= — sin. 6 sin. <p, 

 a / =cos. 6 sin. 9, a=cos. ?, 6"=— sin.d cos. (p, 6'=cos. 6 cos. ?,& 

 - sin. 9, c"=cos. 6 y c'=sin. 4, c=0, (*'). 



i 



