496 
ME. A. COHEN ON THE DIEEEEENTTAL COEEEICIENTS 
So that, using the notation of section 52, we have 
= — 'r;j%{mxz), liy = — ■us%{myz\ \ = m%{m7^). 
If the axis of 2 ; be a principal axis, we have 
'Z{mxz)=0, ^{myz)=(), 
therefore 
/i,= 0, hy=0, 
and consequently H is a line in the direction of O^", and equal to the product of ra- and 
the moment of inertia about 
57. If the axis about which the body rotates is neither a fixed line nor a principal 
axis, it is more convenient to express II, the body’s momentum-couple, in the following 
manner. 
The body’s rotation about the point O may be considered as compounded of motions 
of rotation about the principal axes at O. Let ot,, ■zs-g, zs-g be the angular velocities of those 
component rotations, and let A, B, C be the respective moments of inertia about the 
principal axes. We have shown in the preceding section that the body’s momentum- 
couples due to the three separate rotations about the principal axes would have for their 
respective axes lines along the principal axes and equal to Aotj, Cz^j-g. It follows, 
therefore, from section 54, that H, the body’s momentum-couple, is the resultant of the 
three couples, whose axes are repectively Azitj, Bzzrg, Czhtj. In other words, H, the axis of 
the body’s momentum-couple, has Az^-i, Bzn-g, CziT 3 for its components along the principal 
axes. 
58. The results of the last two sections may be also proved in the following more 
direct manner. 
Take any rectangular axes as axes of coordinates. Let zzy^, n^y, zn-^ be the components, 
along those axes, of the body’s angular velocity of rotation. Let Vy, be the compo- 
nents of the velocity of a particle of mass m, whose coordinates are x, y, z. 
If then liy, be the components of H, we have 
]i,=tm{yv—zvy). 
But •V^=y7n^—X’uyy^ Vy=X'wr^—Zzj^. 
Therefore = zir^m{y'^-\-z^) — ■ujyX{myx) — ■mXi'mxz). 
If, then, the moments of inertia about the axes of x, y, z be denoted by A, B, C respect- 
ively, and if we denote ^[myz) by A', '2[mxz) by B', X{myx) by C, the last equation 
becomes 
7l^ = Azzr^ — ^’’^y — B J 
and similarly, hy=^'my — A'zzr^ — C'Z3-^, 
\ = CzZT^ — B'zzj-^ — A!’UJy. 
These results are true, whatever rectangular axes of coordinates be taken ; but if they 
be principal axes, then A'=0, B'=0, C'=0, and therefore we have, as before, 
ll^-=Kns^^ lly=.'^7Sy^ A^=Cz3-^. 
