498 
ME. A. COHEN ON THE DIEEEEENTIAL COEEEICIENTS 
d'UT 
L+L' = —^{mxz) 
N = 
Mk 
dvT 
dt 
These six equations are those ordinarily given in text-books, and their full import and 
meaning is now apparent. The first three have for them right-hand members the com- 
ponents of D^(U), where U has for its components — 0. 
The last three have for their right-hand members the components of Dj(H), where 
H has for its components —■m%{j)%xz\ — ‘uj'2[myz), And the six equations are at 
once obtained by applying the elementary formulae of section 18. 
60. It is, however, in the solution of problems, far better to avoid using those six 
equations, and simply to remember that the body’s momentum U is Msrr in the direction 
of the velocity of the centre of gravity {r being its distance from the axis), and that the 
body’s momentum-couple Hhas for its components —’UT^{mxz), — ■uj'X[myz), and M^ro-. 
Then the complete differential coefficients of U and H can be found at once according to 
the ordinary rules ; and those complete differential coefficients are by D’Alembeet’s 
principle respectively identical with the force and the axis of the couple to which the 
forces acting on the body may be reduced. 
Take for example the following well-known problem : — 
“ Under what circumstances will there be no pressure on the fixed axis, supposing no 
external forces to act on the body % ” 
Since there are no external forces nor pressures which act on the body, it follows that 
Di(U) and D^(H) must each equal zero. Therefore U and H are lines of constant mag- 
nitude and direction. Now the direction of U is that of the velocity of the centre of 
gravity, and would therefore vary, unless the centre of gravity were at rest. Hence the 
first condition is that the fixed axis passes through the centre of gravity. 
Again, since H is a line of constant length and direction, its components along and 
perpendicular to the fixed axis Oz must be lines of constant length and dii’ection. 
Hence must be constant. Therefore w is constant, or the body revolves with 
uniform angular velocity. 
Moreover the components of H perpendicular to Oz are —'^'%{iinxz)^liy— —7n%{ynyz)\ 
and we have just seen that the resultant of these two components must be a line of 
constant length and direction. But as ts is constant, it is clear that that resultant has 
always the same components along the variable axes of x and y^ and would therefore 
move with the latter, unless those components were always zero. Hence the second 
condition is that X{mxz)=^, 'Z{myz) = ^ in other words, the fixed axis must be a prin- 
cipal axis. It is evident also that the two conditions are sufficient, for they make H and 
U constant lines, and therefore they make Di(H) and D^(U) vanish, and consequently, 
by D’Alembeet’s principle, there are no forces acting on the body. 
