AND DETEEMINANTS OP LINES, 
499 
61. We now come to the case of a body moving about a fixed point O. Let be, at 
time t, the angular velocity about the instantaneous axis, let A, B, C be the moments of 
inertia about the principal axes at O, and let be the components of w along 
these axes. 
H, the axis of the body’s momentum-couple about O, has, we have already seen, for 
its components 
If, then, we denote by O the instantaneous axis, we know, from the fundamental propo- 
sitions in sections 35 and 36, that D,(H) is equivalent to ^ |1 to Ox, ^ |1 to Oy, 
^ II to Oz, together with the determinant of Q to H, O denoting the instantaneous axis ; 
and moreover, that this determinant has for its components 
h.TSy—hy'Gj,, or (C— parallel to Ox ; 
or {K—Oyj^-ur, parallel to Oy ; 
or (B— parallel to Oz. 
Therefore the components of are 
b‘5'+(a-ck»„ 
C‘|f+(B-AK-,- 
But by D’Alembeet’s principle Dj(H) is the same as G, the axis of the couple result- 
ing from the external forces. If, therefore, L, M, N be the components of G, L, M, N 
must be respectively equal to the components of Di(H). Hence we have 
L=A^'+(C-BK^„ 
M=B^+(A-CK»„ 
N=C^^'+(B-AK»^ 
We thus see that these well-known equations of Euler are found at once byresolving 
D((H) along the principal axes, where H is the axis of the body’s momentum-couple 
and has Azsr^, B^-^, for its components, and that they merely express the fact that G, 
the resultant of L, M, N, is identical with Di(H). 
62. The theory of the motion of a body about a fixed point can be more simply inves- 
tigated, and the problems connected with that theory can generally be more easily solved, 
by merely bearing in mind that G=D((H) than by using Euler’s equations, which 
merely express that fact in one 'particular form ; for that form is not always the most 
convenient form, and is in all cases apt to conceal the fact which it embodies. 
