Motion of a System of Bodies. 51 
,B, ,C are each=0: by substituting the values of ce’, c’ ¢ from (0) 
my: 
JV AP? +B? LC +O’?! 
2A ailblbied C | 
VAT EBT LO?” sin. 6 sin. Vara Os (36). (35) 
agree with the formule 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 a’, 
y', 2’, are invariably connected with 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. 
in (35), they become cos. §=—+——= sin. 6cos. = 
dx’ dy! dz! 
In this case 7 ==(0), aa ==()), Gp =O VA—0, B—0, C—O) and 
dp’ 
A, B, &c. are each constant, hence (26) will be changed tong tars 
a“ dN” +. a/dN” + adN’ dq/ NV b/dN’’+B' dN'+bdN" 
a= FE GAO. ar tt ay Roa 
dr’ e/dN+c'dN" + cdN’ 
ap Ted ~ P= rae roe aoe: 
It is evident by (p), that the axes of 2’, y’, 2’ can be found so as to 
satisfy the equations D=Sy/ 2’m=0, E=Sx! 2/m=0, F=Sx/ ym 
=0; then will the axes of «’.y’, 2’, be principal axes. Hence put 
D=0, E=0, F=0; then by (Ul), p'=Ap, q/=Bg, r’=Cr; hence 
Ad “dN + a'dN’+adN’ Bd 
(38), become = +(C —B)q = sal ante eed 
TOU TON b/dN” + bdN’ Cay 
dt uve 
Ce ae 
Bia tell HCE 
Since the position of the axis of « in the plane a, y, is arbitrary, 
we willl now suppose that it makes an infinitely small angle with the 
line of intersection of the planes a, y, and a’, y’, hence neglecting 
infinitely small quantities of the second, &c. orders, we have sin. ) 
=, cos. L=1.".substituting these values of sin. 1, cos. J, in (0), 
we have by neglecting quantities of the order J, a” = —sin. 6 sin. 9, 
a’=cos. 4 sin. », a=cos. 9, 6’ = —sin. 4 cos. 9, b'=cos. 4 cos. ¢,6= 
— sin. 9, c’=cos. 4, e’=sin. 6, c=0, (2’). 
