472 Scientific Proceedings, Royal Dublin Society. 
element ds of the path of the point P, so that in going from 
P to P’ both r and @ increase, the direction of positive rota- 
tion being opposite to that of the hands of a watch. 
At P and P’ erect perpendiculars to OP and OP’ and let them 
meet at Q. Then, if PP’ = ds, we have PM =rd6, and VP’ = dr 
so that the velocity = along PP’ is equivalent to a velocity = 
along PM together with a velocity “ along JP’. Hence, in 
estimating the acceleration along the radius vector we have to 
consider not only the velocity + along 1, but also the velocity 76 
at right angles to it. ‘The former gives an acceleration outwards. 
along 7 measured by - = 7 and the latter gives a centripetal 
acceleration along 7 inwards measured by 7w? = 76%. Conse- 
quently the whole acceleration along * is 
r—1 6 
The advantage of this method, besides its simplicity, lies in its 
bringing into prominence the meaning of the separate terms. 
To obtain the acceleration at right angles to the radius vector 
we have in the direction PM a velocity +6 which gives an 
acceleration in this direction equal to 
further, we have a velocity 7 along IP’ which gives an acceleration 
(v?/o) towards Q measured by’ 
mon “aon 2 «6 dee 
consequently the whole acceleration at right angles to the radius 
vector is 
Pamper gry” ca la ites 
di (70) + 70 Pe a 8) 
7 
which merely expresses that the moment of the acceleration round — 
the origin is equal to the time rate of change of the moment of — 
the velocity. 
1 This may be seen directly to be 76, for the linear velocity along ZP is *, and the 
angular velocity round Q is 0. 
