AOT) DETEEMINANTS OE LINES. 
50S 
D’Alembeet’s principle, since there are no forces acting on the top which have any 
moments about the vertical. Hence we have 
dt 
(€^■2 COS 6—A-!S^ sin =0, 
(I.) 
Therefore 
cos sin 6 is constant. 
Moreover, since two of the principal moments of inertia are equal to one another, it 
follows from section 58 that the sum of the moments of the external forces about O^, 
the a>ris of unequal moment of inertia, is equal to C Hence, as the forces have no 
moment about Oz, we have C^=0. Therefore zsr^ is constant. This relation, together 
CLZ 
with the equation (I.) and the equation of vis viva, solve the problem. We, namely, 
obtain the three equations 
Cz?^ cos 6— Acs ^ sin 
Azy^^- A z3'|+Cz 7^= — 2^5 cos 
where h=Og, and c is some constant. 
66. In some few cases it may be convenient to take moments about lines which are 
fixed in the body but which are not principal axes. Let then H have for its components 
along the rectangular axes of coordinates hy, respectively, then we know, from 
section 53, that 
/i^=Az3 '^— B'z 7^— C'zzr^,, where Al=%{myz), 
Ag=Czzr^— A'zir^— B'zy^, C'=%(myx). 
Therefore, if L, M, N be the moments of the forces about the axes, we have, as before. 
If, on the other hand, the axes of Occ, Oy, Oz are not fixed in the body, but rotate with 
an angular velocity whose components are zc!,, z?r^, zs-^, then we have, in a similar manner, 
L=^+4z?;— &c. 
The above equations are somewhat more general than those given by Liouville in 
his Journal of 1858, and are, as we have just seen, at once obtained by applying the 
fundamental formulae of section 36. 
67. I will now show how the principle of vis viva may be easily proved for a body 
3t2 
