553] two smith’s prize dissertations. 559 
or, what is the same thing, 
Ft 2 F't' 2 
: —, =s : s, 
m m 
that is, 
F' _ m s' /A 2 
F m s \£/ ’ 
and this relation subsisting, and the velocities at the beginnings of the elements of 
time t, t being in the assumed ratio, it is clear that the velocities at the ends of 
these elements of time will be in the same ratio ; and thus the two particles will 
go on moving in the manner in question. 
All that has been said as to two particles, applies without alteration to any two 
systems of particles moving under the like geometrical conditions, and we thus arrive 
at the conclusion; given two similarly constituted systems, which at any instant are 
in a given magnitude-ratio ^ , their component particles being in a given ratio — 
(the same for each pair of component particles), then if the particles of the two 
systems respectively are to move in similar paths of the same magnitude-ratio , the 
s 
times of describing corresponding arcs being in a given constant ratio - (this implying 
7Y it is 
as above that the ratio of the velocities at corresponding points is 
necessary that the forces on corresponding particles in corresponding positions shall act 
in the same directions, and shall be in the constant magnitude-ratio 
s' ' ft'V 
F m ' s ' [tJ ’ 
and this being so, the motion of the two systems will in fact be similar as above 
explained. 
9?î S t' 
Taking —, =/j, for the mass-ratio, -, —a for the length-ratio, and -, =t for the 
m s ° t 
time-ratio ; also , = </> for the force-ratio, the condition determining the force-ratio $ 
is thus 
^ t- 
It is to be observed, that if the forces are entirely internal, and proportional to 
homogeneous functions of the same order, say -n, of the coordinates of all or any of 
the particles; e.g. if they are central forces varying as the inverse ath power of the 
distances; then the condition as to the action of the forces in the two systems 
respectively can always be satisfied by giving a proper constant value to the ratio of 
the absolute forces (or forces at unity of distance); thus, if in the first system we