200 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
pletely expressed by (4.); for, if it be required to find the effect of the superposition 
of two angular velocities represented by co and cJ, we find by (4.) that the velocity 
produced by (w in any point {u) is Da.u, and that produced by J is DoJ .u. The actual 
velocity of {u) will be the resultant of these, that is, 
T)ai).u-\-J)oJ .u, or D{a-{-cJ).u. 
Now by (4.) is the effect of an angular velocity represented by <y-f <y'. 
Hence it follows that the two angular velocities a and cJ superposed produce the same 
effect as the angular velocity co-\-oJ ; and a-\-cJ is the third side of the triangle formed 
of the two lines o) and cJ. 
(103.) The equation of motion of a rigid body acted on by any forces (about its 
centre of gravity) is easily obtained as follows. 
Let U denote the accelerating force in action on m, i. e. at the point («) ; then, 
by Sect. III., we have 
^mu . = ^mu . U, 
which is the same thing (see art. 86) as 
Now, putting for u its value (1.), and supposing that a, /3, y are the p'lncipal axes 
of the body, we find 
2mu.^= (,..^) W+ (/3.f )2my+ 
But, by (4.) and (5.), 
^ = D6^.a= —0^27+ 6/3/3 ; 
Similarly, 
and 
whence, introducing D, we find 
du 
=A6/ia4-Ba/2/3-}-C<a37, 
where A, B and C denote respectively , 
Thus ( 7 .) becomes 
^ (Aft/ia 4- B6/2/3 + C6/37) = SmDw . U, 
(8) 
