502 
ME. A. COHEN ON THE DIEEEEENTIAL COEEEICIENTS 
Therefore, if L be the moments of inertia of the forces about the principal axes, we 
have by D’Alembeet’s principle, 
L = ^ 
~ Aro-^OT^. 
It is clear that, when there is more than one set of principal axes at the fixed point, 
either all three or at least two of the quantities A, B, C must be equal to one another. 
Suppose then A=B, then the axis of 2 , O 2 infixed in the body, and therefore andw^ 
are clearly the same as vs^ and And the last equations, therefore, become 
L=A^^+t.,(G^.-A<), 
M=A ^+cr^(ATO-' — Cot^), 
dt 
d/Y 
If we put ■57^=®-^-}-^^, the above equations become the same as those which are given 
in Routh’s ‘Dynamics,’ page 134, where they are deduced from Euler’s equations. 
65. To the above equations, however, the same remark applies as has already been 
made with regard to Euler’s equations. They merely express the fact that 0^(11) has 
L, M, N for its components ; and it is far better in most problems to start with that 
simple fact, and, without using those equations, to choose any axes which the nature of 
the problem may suggest. 
Take, for instance, the problem of the top spinning upon a per- 
fectly rough plane. 
Let O be the fixed point, g the top’s centre of gravity. Take O g 
as axis of z. Draw O a vertically, and take as axis of a? a line per- 
pendicular to O 5 ; and in the plane zOa^ and take as axis of ^ a line 
perpendicular on the plane zOx^ and therefore perpendicular on O g. 
The axes of coordinates are evidently principal axes. 
The components of H, which determines the body’s momentum-couple, are Atsr^, Bcy^, 
OTg being the angular velocities about the principal axes, and A being the 
moment of inertia about O x and about O y, and C being the moment of inertia about O g. 
We have chosen O ^ so as to be perpendicular on plane aO X', consequently the resolved 
part of H along O a is the sum of the resolved parts of Cw*, aud equals therefore, if 
we denote angle a O 2 by 5, 
— Aro-^ sin cos &. 
Moreover, since O a has a fixed direction, the differential coefficient of the last expres- 
sion is evidently the resolved part along O a of Dj(II). But this must equal zero by 
