52 Motion of a System of Bodies. 



Multiply (39) by dt, substitute the values'of a" s a', &c. from (a/), 

 and we have Adp + (C — B)qrdt= — (s\n.6 rfN'"-cos.d dN") sin. 

 p+cos. (p </N', B<fy + (A - C) prtf< = - (sin. 5 </N"' - cos- 6 dN") 

 cos. cp -sin. 9 rfN', C</r+(B-A) pqdi^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 formulas are equally applicable, 

 whether we consider the motion of a rigid system, or solid, about a 



fixed point. 



Again, x\ y\ z\ being principal axes, (r') will be changed to T 



Aji'+Bj'+Cr* ; ; (dT\ m (dT\ (dT\ % 



. # .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 + fB} + c"Cr 



A 7 , a'Ap + b'Bq+c'Cr^B', aAp + bBq + cCr^C, (41), A 2 p 2 

 +B 2 g 2 +C 2 r 3 =A /2 + B /2 + C /2 , (42); also (39) become-— -f 



(C-B)qr=0, -J+(A-C)pr=0, ~ +(B- A)pq=0, (43); 



multiply (43) by p, q, r, severally, add the products, take the integral, 

 and we have A/> 2 + Bg 2 +Cr 2 =D'=const. (44), which agrees with 

 the well known principle of the preservation of living forces, multiply 

 (43), by A/?, Bg, Cr, then proceed as before, and we have A 2 p 2 -f- 

 B 2 9 2 + C J r 2 =E /2 =const. (45); this compared with (42) gives 



A' 2 +B' 2 + C /2 =E' 2 , (*')• 



It is evident that the system may be considered as having a mo- 

 mentary axis of rotation : to find the momentary axis we observe 



that relative to it, we shall bave-,-~0, ^=0, -r = 0,.\by (g'), 



L=0, M=0, N=0, or by (/),(and because ~ =0, 77 = 0, -^ =0,) 



we have qz' — n/'=0, rx' - ps'=0, py' - qx' = 0, (46) ; which indi- 

 cate that the momentary axis is a right line which passes through the 

 origin of the coordinates. Let a n b n c, respectively, denote the co- 

 sines of the angles which the momentary axis makes with the axes of 



V 



p 2 +q*+r* vV* +y s +z* 



a 9_ , 3 ^_ , r 



</p* + ? 2 + r 2 VV 3 +/*+*'* " Vp* -f q 2 ~+ r 



a 



xf 



V^T«>T^- C " (47) " 



