52 Motion of a System of Bodies. 
Multiply (39) by dt, substitute the values’of a”, a’, &c. from (zx’), 
and we have Adp+(C—B)qr dt= —(sin. 6 dN” — cos. 6 dN”) sin. 
etcos.¢ dN’, Bdg + (A—C) prdt= —(sin. 6 dN” -- cos. 6 dN”) 
cos. 9 —sin. 9 dN’, Cdr+(B—A) pqdt=cos. 6 dN’’+sin. 6 dN”, 
(40); which agree with (D) given at p. 74, Vol. I. of the Mecanique 
Celeste, for the motion of a solid about a fixed point; as they evi- 
dently ought to do: for the above formule are equally applicable, 
whetlier we consider the motion of a rigid system, or solid, about a 
fixed point. 
Again, 2’, y’, 2’, being principal axes, (r’) will be changed to T= 
Ap?+Bq?+Cr? dT dT dT 
ror i ee henco| a Tee Ge a) =Ba( =Cr; 
.*.in the case of (30) and (31), (that is when the system is not af- 
fected by any disturbing forces,) we shall have a’”Ap+b’Bg+-c’Cr 
=A’, a’Ap+b/Bq+e’Cr=B’, aAp+bBq+cCr=C’, (41), A’p? 
Ad, 
+B?%q?2+C?r? =AZ?2+B2+ Cz, (42); also (39) become —+ 
Bdq Cdr 
(C—B)qr=0, 7, +(A-C)pr=0, 7, +(B—A)pq=0, (43) 5 
multiply (43) by p, q, 7, severally, add the products, take the integral, 
and we have Ap? + Bg? +Cr? =D/=const. (44), which agrees with 
the well known principle of the preservation of living forces, multiply 
(43), by Ap, Bg, Cr, then proceed as before, and we have A?p? + 
B2q?+C?r?=E’? =const. (45); this compared with (42) gives 
A?+B?24C”? aa i0/2., Hi) 
It is evident that the system may be considered as having a mo- 
mentary axis of rotation: to find the momentary axis we Tusa 
{ : dx dy dz 
that relative to it, we shall have77=0, 7, =0, i= 8” Dy AGoa)s 
dar! dy! dz’ 
L=0, M=0, N=9, or by (f’),(and because aa) a 0,) 
we have qz’—ry’=0, rx! — pr! =0, py’ — qx’ =0, (46); which indi- 
cate that the momentary axis is a right line which passes through the 
origin of the cordinates. Let a,, 6,, ¢, respectively, denote the co- 
sines of the angles which the momentary axis makes with the axes of 
a! 
x’, y’, 2’ respectively, then by (19) ess — pp? Tas hh) yen Ly prey 
2 x? y 2 
q y" r 
=a, ———————— = ———S Ss b, pe ea 
TY pee qh ee Ne ee ee Op? ee 
pall 
=c, (47). 
Va py? pa 
