AND DETEKMINANTS OE LINES. 
501 
Let, as before, G be the resultant of L, M, N, and let H be the axis of the body’s 
momentum-couple about the fixed point, and let O represent the' instantaneous axis. 
Then G=D((H), and ^^(H) is equivalent to [[ to Ox, II to Oy, C ^ |1 to Oz, 
together with det (O, H). 
Now, if A=B=C, then H, which has for its components Czs-^, evidently coin- 
cides in direction with O, which has for its components Therefore it follows, 
from the very definition of a determinant, that det (O, H)=0. It is therefore because 
det (O, H)=0 when A=B=C, that the components of ©^(H) are simply A B 
CiC ttC 
C and that Euler’s equations take so simple a form. 
dt 
Secondly, suppose only A=B. It is evident, from what has been just said, that 
N=C — +the resolved part along Oz of det (O, H). Now the equation to the line H is 
dt 
X y z 
A'STx BOTj, 
If then A=B, the projection of H on the plane of xy evidently coincides with the 
projection of O on that plane. Therefore the lines H, O, and the axis of z He in the 
same plane. Hence it follows that the Hne det (O, H), which by definition is perpendi- 
cular on Q and on H, is also perpendicular on the axis of z, and has therefore no com- 
ponent along that axis. We thus see that the reason why N=C^, when A=B, is 
that in that case det (O, H) is perpendicular to the axis of z. 
64. In those cases in which there are more sets than one of principal axes at the fixed 
point, it is sometimes convenient to take moments about a set of principal axes, which 
are not fixed in the body. 
There is no ditficulty in applying the same method to such cases. Let ro-^, be 
the angular velocities of the body about the principal axes O^, Oy, Oz, and suppose those 
axes not to move with the body as if rigidly connected with it, but to move at time t about 
an instantaneous axis O' Avith an angular velocity equal to the length of O', and let the 
components of O' along the principal axes be 
Let H, as before, represent the body’s momentum-couple. We have seen that H has 
for its components 
= Acr^, = Bot^, \ = Cro-^. 
Therefore, according to the fundamental proposition in section 35, is equivalent 
to ^ (AwJ [1 to Ox, ^ (Bnr^) II to Oy, ^ (CvT^) II to Oz, togetherwith det(0', H); and looking 
nt the formulae of section 36, we see that det (O', H) has for its components 
parallel to Ox, 
parallel to Oy, 
— Azir^r^^ parallel to O’:, 
3 Y 
MDCCCLXII. 
