16 
On the Method of demonstrating some Propositions in 
Dynamics. By I. Topuuntsr, M.A., F.R.S., St John’s 
College. 
SUPPOSE a particle moving in a straight line; let s be the 
Space described at the end of the time ¢, v the velocity, f the 
acceleration; then we have the equations 
dv d’s 
if cg f= Ge: 
And similar equations hold in more complex cases of motion. 
Thus we have theoretically the choice of two methods when 
we wish to determine /; namely we may first find the velocity 
and then f from the relation f= = ; or we may find / without 
2 
attending to the velocity from the equation f= EE 
The propositions given in Newton’s second and third sec- 
tions are in effect treated in the latter of the two methods. It 
is however quite possible to treat them by the former method ; 
and the following advantages seem to follow by adopting the 
first method. 
The results can be obtained without requirmg so much 
knowledge of the properties of Conic Sections. é 
The theory of limits is used in a more simple and convincing 
manner. 
The illustration of mechanical principles is more varied. 
As an example take the most important proposition, namely 
that of motion in an ellipse round the focus. 
The figure may be easily constructed. Let S be the focus 
of force, H the other focus, P and Q two points on the ellipse. 
It may be shewn in the most elementary manner that the 
