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doubt, the diagram and the process of application of the 
principles admit of greater simplicity even than that which 
has been given in the paper. To this end it is necessary to 
state that the principle and formula of the conservation of 
energy are given by the simple dynamic of constant forces 
in such wise as to be comprehended without the aid of 
differential equations. 
The definition of work is simple, but, the estimation of it 
when the force is variable depends upon quadratures. 
When, however, the force varies as the distance the esti- 
mation of the work done by it is given by Moseley, Twisden, 
and others by elementary mathematics, and is \ px 2 , using 
the notation as given in the paper. 
Taking, therefore, for granted a knowledge of the principle 
and formula for the conservation of energy, the formula for 
estimating work done by a force which varies as the dis- 
tance from a given point, the resolution and decomposition 
of forces — the diagram and solution of this problem, will 
stand as follows. 
Let 0 be the position of a body, 
whose weight is (w;) and velocity (%), 
moving towards A in the straight line 
BOA where OA = OB. Let {p) represent 
the force at a unit of distance from 
O, then (px) will be the force at P, 
where OP = x. Describe the circle about 
O as centre with the radius OA = a, and 
draw PQ perpendicular to AB meeting 
the circle in Q. Put (y) the velocity of (w) at P. 
Now the work done against the force {px) in moving 
from O to P must be equal to the accumulated work lost by 
passing from (v 0 ) to (v). Therefore 
w 
= ( 1 ) 
When, however, v = 0, then x = a, and the weight (w) is at 
A. On this supposition (1) becomes 
