OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
199 
This result is of great importance, and furnishes, in the simplest possible manner, 
every formula necessary for determining’ the motion of the rigid body. 
(98.) The symbol a represents, in direction, the instantaneous axis of rotation. For 
du 
( 2 .) shows that, when u coincides in direction with <y, -^^O; consequently all the 
points of the body which lie in the direction of u are quiescent at the instant t. 
( 99 .) The magnitude of a is the instantaneous angular velocity. For, let a' denote any 
dc^ 
unit line fixed in the body at right angles to u\ then (see art. 19) is the angular 
dai 
velocity, at least in magnitude. But, by ( 2 .), -^=D 4 ;.a' ; and, since a is at right 
angles to a', D^y.a' has the same magnitude as a (see art. 40.). Wherefore a has the 
da! 
same magnitude as -gp, and therefore represents the angular velocity in magnitude. 
( 100 .) Hence the result above obtained, namely. 
u. 
(4.) 
may be thus enunciated : — the rigid body is, at the time t, moving about a certain 
instantaneous axis, with a certain angular velocity ; and if we assume a to denote 
that axis, in direction, and the angular velocity in magnitude, then the velocity 
of any point (w) of the rigid body is obtained by performing the operation (Du.) 
upon u. 
( 101 .) If we put 
u=ct)iCc-\-cii). 2 j^-\-U 3 'y, (5.) 
where Ui, denote numerically the projections of the line u on the three coordinate 
directions a, (3, y, we find by (4,), 
dvL 
-g^=co(Da.u-\-uf)^.u-\-uJi)y .u (6.) 
Hence it appears that ^ results from the superposition of three angular velocities 
Ui, Ui, U 3 about the axes a, j3, y respectively ; for, by (4.), D(u^a,) .u (or u(Da.u) denotes a 
velocity of the point (u) resulting from an angular velocity about the axis «, 
that resulting from about j3, and af^y.u that resulting from about 7 : 
and ( 6 .) shows that the actual velocity of u is the resultant of these three velocities. 
If we put for u its value xci-\-yl3-\-zy, ( 6 .) becomes immediately 
du 
Whence it follows that the velocity of the point (xyz) is equivalent to the three com- 
ponent velocities a^z—a^y parallel to x, u^x — a^z parallel to y, and a^y—u^x parallel to z. 
( 102 .) The theory of the composition and resolution of angular velocities is com- 
