MOTION. 
before, they must continue so ; but the body 
F, having no other body to re act upon it, 
has nothing to obstruct its motion ; it will, 
therefore, move on with the same velocity 
which A had at first, because it has all the 
motion of A, and the same quantity of mat- 
ter by hypothesis. 
Let there be three bodies, A, B, C, (fig. 
11.) and let A strike B at rest j the velocity 
generated in B by the stroke will he y = 
^ ® and so the momentum of B will be 
Q + Q’ 
2QVQ 
=zQy. With this momentum B 
Q + Q 
will strike C at rest and contiguous to it ; the 
2 Q ?/ 
velocity generated in C will be ^ , ■ 
, ^QyC 2QC „ 
its momentum wdl be rr-| — rr= >. i X 
; and 
Q + C 
4QVQC 
■Q + C 
QQ + aC-f Q^ + QC 
2QV 
Q Q 
If now we suppose B a variable quantity, 
while A and C remain the same, we shall 
find what proportion it must have to each 
of them, in order that the momentum of C 
may be a maximum, or the greatest possi- 
ble, by putting the fluxion thereof equal to 
. 4Q^C^VQ — 4QCQ^Q 
QC + QQ + QC -fQ^I^ 
= 0 ; whence we get QC — Q Q =z 0, and 
so Q C = Q Q , consequently Q : Q Q : C, 
or A : B :: B : C ; that is, the body B is a 
geometrical mean between A and C. Hence 
if there be any number (n) of bodies in a 
geometrical ratio (r) to each other, and the 
first be A ; the second will be r A, the third 
A, apd so on to the last, which will be 
r”-' A. 
Also, the velocity of the first being V, 
2 V 2 Q V 
that of the second will be ^ (for 
2A V 2 V 
is here = 
tliird 
4V 
A-frA 
, that of the fourth 
that of the 
8V 
1 -|- r 
-5— ).-i 
, and 
1 + r' 
so on to the last, which will be — f — i V. 
1 -4- r 
The momentum of the first will be A V, 
that of the second — , that of the third 
1 + ’■ ’ 
^ fourth — 
1+/’ l+r 
en to the last, which will be AV. 
To give an example of this theorem ; if 
n = 100, and r = 2, then will the first body 
A be the last r”— ‘ A, as 1 to 
33825308000000000000000000000, nearly ; 
A V 
— , and so 
and its velocity to that of the last nearly as 
271022000000000000 to 1 : lastly, the mo- 
mentum of the first to that of the last will 
be nearly as 1 to 2338480000000. 
If the number (n) of bodies be required, 
and the ratio of the momenta of the first 
and last be given as 1 to M, and the ratio 
of the series r given also ; then, putting 
— R we have the momentum of tlie 
1 -fr ’ 
last body expressed by ^ = M = 
R"— « ; therefore the logarithm of M (1. M) 
is equal to the logarithm of R (2. R) multi- 
plied bv the power n — 1 : that is, 2. M = 
n — 1 X 2. R ; consequently 
2.M 
2.R. 
-1-1 — n, 
the number of bodies required. 
Motion, in botany, implies not so much 
a change of place as a change of direction. 
The direction of the roots and stems of 
plants is totally opposite, the former either 
running directly downwards, or extending 
themselves transversely or horizontally Tin- 
der the surface of the earth ; the latter ex- 
hibiting motions of a similar nature, but in 
a contrary direction. The direction of the 
root is never vertical, except in the sanar of 
Senegal, the roots of which twisting, rise 
vertically upwards a foot above the surface 
of the earth, and ai'e sometimes covered by 
the flux of the sea. Familiar as the appear- 
ance is, naturalists are not agreed with res- 
pect to the causes which determine the 
roots of plants to tend universally down- 
wards, either in a horizontal or perpendicu- 
lar direction ; and the stems, on the con- 
trary, to mount perpendicularly or horizon- 
tally upwards. So constant, however, are 
these opposite directions, that a plant being 
taken out of the earth, and replanted in it in 
such a manner that the root is uppermost, 
and the stern lowermost ; the root will 
quickly curve downwards, the stem up- 
wards, till each has resumed the direction 
which is proper and natural to itself. 
All the causes which concur in promoting 
the growth of plants appear likewise to 
operate in determining their direction. Such 
are the air, the sun, light, and the moist 
warm vapours which arise out of the earth. 
The three first seem to concur most cer- 
tainly to the direction of the stem ; air and 
moisture to that of the root. If any num- 
ber of plants arc placed in pots, in a room 
which only admits the light by a single hole, 
the stems will incline or direct themselves 
towards that side. In thick forests, the 
young trees always lean to the side where 
