FOR 
general equations, which contain nae principles or theorems 
known under 
onfervation o 
Confervation of the eatin oy the centre of gravity, 
Confervation of equal areas, 
ed the principle of thie leaft actio 
_ The firft of thefe principles, the confervation of the 
he was 5 dileovere y re sy but under a form ee 
erent to that which we now give it. n€ prin- 
ene, as employed by him, confit in the equality between 
the — ent [an defcent of the ¢ tre of gravity of feveral 
deferiied in the fame direCtion, ‘divided by the fum of che 
e other hand, by the theorems of Galileo, 
the vertical face defcribed by a heavy body in its defcent 
is proportional to the {quare of the velocity acquired, and 
with which it will afcend to the fame height: thus the 
n any manner 
whatever, or have each defcended freely gk the fam 
from | the action yf 
wack to the fimple vis viva ay egs: 
le gave it oa 
the a€tual forces which m 
name of the ¢€ een seer He vi and employ 
it with fuecefs in the folution a ead pete that dee 
him had not bee 
from this principle ae aw of hee motion of flu pe a 
a fubje&t which before had only been ne ted ina vague 
and. nfatisfa@ory m ner. In the Berlin memoirs for 
1748, he explained aa rendered this principle very general, 
fhewing how. it might be applied to the motion of bodies 
folicited by mutual attraction, or drawn towards fixed 
centres by forces — to any funtion of the dif- 
tance whatever. 
variable quantities are reduced to one, this auation alone 
is fufficient for its spay which is the cafe in that relating 
to ie — of a lation. In general the confervation o 
the @ gives always a firft integral of the different 
differential. saan one of each problem, which is often of 
reat utility. 
< The feeond principle is that of Newton, given as an ele- 
mentary ‘propofition in his “‘ Principia.’”? He demonttrates, 
that the ftate of repofe or motion of the centre of gravity 
. of feveral bodies, i is not altered by the reciprocal action of 
thefe bodies. in any manner whatever; fo that. the centre 
CE. 
gravity of bodies which aét upon each other either by- 
ords or levers, or by the laws of attra€tion, independent of* 
any exterior aétion or obftacle, eine always in repofe, or 
moves uniformly in a ftraight li 
D’Alembert has een is this theorem, and fhewn that 
it migh t 
of this centre will be the fame as if all the hi of the bodies 
were applied to iteach inits proper dire 
It is evident that this principle eee na determine the 
motion of the centre of gravity, independently of the 
refpective motions of the bodies, and thus it will always 
afford three finite equations between the co-ordinates- 
of the bodies and the times, and which will be the inte- 
grals of the differential equations of the problem 
he third principle is much lefs ancient than the two 
ieee and appears to have been difcovered about the 
fame time by Euler, Bernouilli, and the chevalier d’Arcy, 
but oat different forms. 
According to the two firft, the principle confifts in this, 
that in the motion of (eid) bodies a 
e velocity of rotation round the centre, and it 
a the nee soak is always independent t of any mutual 
on w th ies may exert upon eac h other, and 
prefernes fel is Gate as long as there is no exterior a¢tion 
obitac Daniel Be paula gave this paar in the. 
ff saree of the Memoirs of the Academy of Berlin,-. 
in 1746, and d’Alembert the fame year, in he “ ‘Opuleulay? 
The principle of M. d’Arcy, as given to the Academy of 
Paris in 1746, but not printed till 1752, is, that the fum of 
the produts of the mafs of each body, by the area traced by | 
its radius vector about a fixed point, is always proportional © 
to the times. | This principle 1 is nothing more than a gene- - 
ralization ewton, ote de Sere ” 
of areas delovited by centripetal ie and to he 
analogy, or rather identity, with that of Euler and “Bernouilli : 
it is fufficient to recelle&t that the velocity of rotation is ex- 
ent of the d ab 
laft principle is only the diff 
M.d’Arcy. This author afterwards gave this principle . 
another form, which renders it more fimilar to the preceding. 
The fum of the produdts of the maffes-by the velocities, . 
and by the perpendiculars drawn from the ol to the - 
~~ of the forces, is always a conftant quantit 
er this point of view he eftablifhed a Gee of meta- 
laws of mechanics'to the rank o fe 
But however rae may be, the oanagk | in : quefion takes - 
place | in 7 fyfte of raid es which a h other in 
an 
a ree ce and mo 
toa centre; and this, whether the fyftem be ra free, 
or conftrained to move about this centre. ‘The fum of the 
produdts of the mafles by the areas defcribed about this 
centre, and projected on any plane whatever, is always pro- 
portional to the time; fo that referring thefe areas to three 
reCtangular 
