Chase. ] 396 [March 1, 
proof as soon as possible, must be admitted, and I now submit some 
further considerations, which, in my own judgment, impart a more strictly 
mathematical character to my fundamental postulate. 
It is well known that the velocity acquired in falling towards an 
attractive centre, depends upon the attracting mass and the distance 
fallen through. In other words, 
Rh Bh im 
VXV 2h XV @ d \2h 
It is moreover evident that in any perfectly elastic particle, oscillating 
perpetually in a compound orbit, about two centres of attraction, as in 
the hypothetical case of water vapor, set in motion by the force of 
chemical combination, if ‘ 
mu & 2h, 
the proportionate velocity communicated by gravity varying as Va’ the 
d 
proportionate living force will vary as The mean amounts of living 
2h 
d- 
force imparted by each of the two attracting centres will then tend con- 
stantly to equality, thus counteracting any indefinite expansion or con- 
traction towards the centre ‘of prepondering attraction, which would 
otherwise gradually draw the oscillating body to itself. This exigency 
can be satisfied, and a perpetual oscillation maintained by the conjoint 
action of gravity and elasticity, only when 2/ has the proportional value 
here indicated. 
The question may be approached in another way. The sustaining vis 
vivog of the earth in its orbit, having been acquired by a virtual fall through 
i 
) 
a 
the half-radius Be i, let it be required to find the proportionate part 
of the possible fall which will sustain the elastic oscillation. 
Since the attracting forces (or the virtual masses acting at the point of 
. Wd . . 7 ~ . Vb 
disturbance), vary as a) the virtual centrifugal forces will vary as i" 
a" ¢ 
Then iy keep? 
Me nl a ae 
Wend 
Nal galt nai 
OFI) m''d!  ™m nN" 9 
210 Se $m & 2h & a. 
ae 
m! 2h a! 
At whatever distance from the centre the elastic particle encounters an 
obstacle, a portion of the force must be communicated to the obstacle, 
originating new molecular motions, which, if they could all be known, 
would show that the aggregate amount of force is still maintained. The 
following attempt to trace a portion of the transmitted forces of inter- 
