470 Scientific Proceedings, Royal Dublin Society. 
give a resultant of the same magnitude as P but differing in 
direction. The effect of Q in general, therefore, is to change the 
magnitude and direction of P; but it may be so chosen as to alter 
either of these and leave the other unchanged. 
It is this principle of compounding and resolving according to 
the parallelogram law that underlies the whole science of Mechanies, 
and whenever we employ it we deal with the problem more nearly 
from first principles, and therefore keep in view more prominently 
the nature of the processes and assumptions by which we arrive at 
our final result. 
As a first illustration of this principle of resolution, let it be 
required to write down the component velocities of a point parallel 
to the axes of reference when it is given that the point describes a 
circle round the origin with uniform 
angularvelocity. In thiscase the velocity 
of the point is perpendicular to the 
radius vector OP (fig..2), and propor- Oe 
tional to it being equal to wr, and 
therefore by the principle of resolution ii. 
this may be replaced by a component 
velocity perpendicular to v, and pro- 
portional to #, together with a com- 
ponent perpendicular to y and propor- 
tional to y. That is, the velocity wr perpendicular to 7 gives 
components wa perpendicular to zw, and wy perpendicular to y. 
Mi 
Oo Oy x 
Fig. 2. 
— eee ee 
Hence, for the direction of rotation opposite to the hands of a — 
watch the component velocities in the directions of the axes of — 
reference are, obviously, 
U=—- WY, V= wi. 
We shall now employ the same method to determine the accele- | 
ration of a point P which describes a circle 
with uniform angular velocity. Let O be of 
the centre of the circle, and P’ a position 
of the point so close to P, that the are PP’ o ei 
may be supposed to sensibly coincide with 
its chord. Then, since the velocity at P Fie. 3. 
is perpendicular to OP, and proportional to it, viz. w7, and since the — 
velocity at P’ is perpendicular to OP’ and proportional to it (viz. wr 
