66 Le Roy’s Memoir on the best Method 
this point, if it had only consumed the velocity twenty in- 
stead of that of thirty; in the penultima it has the necessary 
velocity to overcome the resistance of the accelerating force; 
in the last it has besides, that of friction, or of the air, which 
it experiences there; in the antepenultima it has the requi- 
site velocity to surmount the resistance of that same force 
in the two last, plus that of the air, friction, &c. Applying 
the same reasoning to all the other points of the space 
twenty, we shall find that in the present case the velocity 
there is greater than if, disengaged from every obstacle fo- 
reign to the accelerating force, A should describe the space 
twenty only, by means of the initial velocity twenty: the 
same must be concluded for every other space. Therefore 
the foreign resistances experienced by a body in vibration 
shorten the time of the retarded semi-vibrations. 
Corollary 1. 
The inverse of the preceding evidently takes place in the 
accelerated semi-vibrations. 
Corollary II. 
The obstacles which may be opposed to the motion of 
a body in its retarded semi-vibration, being so many causes 
which make it stop the sooner; on the contrary, in the 
accelerated semi-vibrations, the foreign obstacles destroying 
a part of the acceleration, and hindering this semi-vibration 
from being made so readily, (since every body which oscil- 
lates necessarily feels some foreign resistances, whether from 
the air, from the friction of the parts which sustain it, or 
from the particles themselves of the spring which holds it;) 
it follows, that in every body which vibrates, the retarded 
semi-vibration is always quicker than the accelerated semi- 
vibration which succeeds it; and that the more considerable, 
the resistances are which we have just spoken of, the greater 
is the difference between the time of the acceleration and the 
retardation. 
Ol servation ¥. 
In the vibrations that a body makes by the help of an 
elastic force, or of gravity, if we admit that it is the effect 
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