500 
ME. A. COHEN ON THE DIEEEEENTIAE COEEEICIENTS 
Take, for instance, the problem of a body rotating about a fixed point, no external 
forces acting on it. Here r)^(H) = 0, therefore H is a line of constant length and direc- 
tion. The motion of the body must therefore entirely depend upon the fact that, whilst 
the body moves about the princijpal axes ivith angular velocities the lineH, whose 
projections on those axes are respectively Bra-y, remains throughout the body's 
motion the same in magnitude and direction. 
The length of H is evidently But the length of H is constant, 
say equal to h. Therefore 
AV^+BV^-|-CV^=A^ (I.) 
Moreover, from section 35, we see that H;(H) is equivalent to A^ [| to OiZ?, 
B ^ II to Oy, C II to Oz, together with det (H, H) ; and since the last line det (Q, H) 
is perpendicular on the instantaneous axis O, it follows that the resolved part of T)^(H) 
along the instantaneous axis equals 
^ A I B I ^ C 
dt 'GT dt 'GT dt 
But this must equal zero, since Hj(H) equals zero, and since consequently its resolved 
part along any line is zero. Hence we have 
Therefore 
AOT^+BOTy-f-CziT^ is constant, equal, say, to (II.) 
We have already seen that H is a line of constant direction; and since its dfiection- 
cosines are proportional to Azb-^, Bt^j-^, Czt^, it follows that the plane whose moment has 
direction-cosines which are proportional to the last three quantities is a fixed plane. 
This plane is the invariable plane. From this fact and the two equations (I.) and (II.), 
Poixsot’s celebrated illustration of the motion of a body which rotates about a fixed 
point may be easily deduced in the ordinary manner ; but it is unnecessary to discuss 
the problem further, as it must be already sufficiently apparent that the body’s motion 
entirely depends upon the fact that H is a line of fixed length and direction. 
63. On looldng at Eulee’s equations, we find that, when A=B=C, they take the 
simple form 
M=B 
dz7,. 
N=C 
d'^z 
dt 
Moreover, when only A=B, then the third of Eulee’s equations becomes N=C 
It may be interesting to trace the real meaning of these results. 
r'! 
dt ' 
