AND DETEEMINANTS OE LINES. 
497 
59. We are now in a condition to solve easily the problem of the motion of a body 
rotating about a fixed line, or about a fixed point under the action of any forces. 
First, let us take the case of a body of mass M revolving about a fixed line. Take 
that line as axis of s, and choose for the axes of x and y any \me^ fixed in the body 
which are perpendicular to one another and to O^. 
It was proved in section 52 that H, the axis of the body’s momentum-couple, has for 
its components 
h^——w%[mxz\ liy=—'us%{myz), ( 1 .) 
standing for the moment of inertia about Oz. 
Moreover we know from section 48 that U, the body’s momentum, is MV, where V 
is the velocity of the centre of gravity. Now if x, y, z be the coordinates of the centre 
of gravity, Y has evidently for its projections on Ox, Oy, Oz, — zs-x, 0 respectively. 
Hence the components of U are equal to 
M^y, Uy='M.zux, (Il.j 
knowing, then, the components of U and H, we can easily find the components of I);(U) 
and D^(H). Using the notation of section 52, P=:D^(U) and G=D^(H), and the com- 
ponents of P and G may be denoted by P^, P^, P^, and G^, G^, G^ respectively. 
In the problem now before us, the axis of z does not move ; hence evidently 
(III.) 
But as to the axes of x and y, they revolve about the axis of z with an angular velocity 
w at time t. Hence by the elementary formulse of section 18 in Chapter I., we have 
Similarly, (IV.) 
^X=jfiK)—hy-Uy, Gy~{hy)-{-h,7U. ^ 
Let, then, external forces acting on the body be reduced to a force at O whose compo- 
nents are X, Y, Z, and to a couple the components of whose axis are L, M, N. Let 
the reactions of the fixed axis be similarly reduced to a force whose components are 
X', Y', 7J, and to a couple the components of whose axis are L', M'. Then, by D’Alem- 
beet’s principle, 
X-|-X'=P^, &c., 
L-1-L'=G^, &c. 
Therefore, substituting the values (I.) and (II.) in equations (HI.) and (IV.), we obtain 
the following six equations : — 
X+X'= — 'M.y 
Y+Y'= 
Z+Z'= 0, 
