FORCE, 
reCtangular planes, tre differential equations are obtained 
of the firft order between the Bike and the co-o1 etnates of 
ke 
Maupertuis, and which the writings of feveral illuflrious A 
thors have fiuce rendered celebrated? Confidered analytically 
ie ape is eae thatin ay pred of bodies which a& one 
» the fum e produdts of the mafles by ¢ 
ie wat ce aa by thé ee defer:bed, is a minimum. 
The author has deduced from it the iw of refleGtion an 
refraGtion of light, as likewife thofe of the fhock of badics, 
in two memoirs, one of the. cademy of Sciences for 1744, 
the a two hia afterwards, in thofe of Berlin 
m onfeffed, ee ae ce are to 
ou _ eftablith ie truth neral principle, nas 
befides, have fomething in a too vagu e and en 
which renders the confequences uncertain as to the exa¢t- 
nefs of the principle itfelf: fo that this — ought not 
to be clafled with thofe above explained... But 
other point of view in whic 
curve is alwa a 
This property, as le ee ave was pnt ae wn by 
Euler to belong to infulated bodies; La Grange extende 
it oa ie to on SS of bodies which an on each 
he produéts of the maifes 
of-the vis viva, a folution of many difficult problems‘in-dy- 
namics may be-obtained. -Examples are to be found by La 
Grange in the fecond volume of the-Menioirs of the Aca- 
demy of Turin. 
Such is the -general_ a of the fee and na- 
ture of -the- ao amics,* as given by » 
Grange... The ie fubject ein ta uch in the cae manner 
by La Plaée in the “ Mechavigne Celette,” oo the inveftiga- 
fur La G: 
sn i is carried {till «Wi th 1e adopts the 
ting the principe of virtual velocity to be aflumed as a fun 
damental axiom, but demonftrates it by a regular - n of 
induétions 5 having ¢€ eftablifhed nearly the fame formu 
difecenval equations, and. deduced all the-above vseneal ous ‘ 
ciples in the manner already: defcribed. 
In addition to ih principles ‘others, in the nature of - 
a a introduced, many of which are very deferv- 
ing of att , From the PocaDe a confervation of 
hed bodies fo- 
pie of the wis viva, 
‘centre of gravity, even en fuppofing it to have a reCtilinear. 
range in not admit- * 
calcitlus, we fha 
and uniform oe $ it follows, 
mined paffing th 
fum of the eee deen 
that a oe may be deter- 
moveable origin h the 
a 
a 
3 
= 
5 
e 
a 
PS) 
a 
QO 
o2 * 
Se 
a, 
= 
— $c 
ct 
o 
o> 
"8 
ve] 
La Plae ce roceeds to examine in what manner thefe re- 
fults would cs changed, by Lea other relations to fub- * 
fift heard the force and veloc Force 
infinite number of ae ie iv ely t 
belides we fimple law of proportio tie 
any mathematical contradi¢tion, 
to be fome other fun&io on of the v 
s mafs," oe the’ double oF 
the integral of its velocity, sruleiplied by the differential of 
the funtion of the veloc city w 
t is in the law t th Ye motion or the 
centre of grat is pity uno and reCtilinear, 
to every poffible’ relation ‘between force and veloc 
This principle of the leaft a@ioil is not fo obvious as others 
we’have mentioned, ' being much more remote from the ele- 
an he oe from which they cen nae d; butif it 
The faQ, low ver, i$ én ea. deferving of atten- 
tion; it-may be vanadyl flared thus: fuppofe a a 
point, in confequence | of the action of fever a to 
n the pat rve defcribe dis is 
found te have this remarkable property, that the integral or 
duct of the velocity (a 
ae ur to illuttr ate it t by takin a par- 
ticularcafe as an exa I M (fe +2.) 
projected Gee zonclh in the direétion M s, and at the fame 
ae ‘attracted by the force of gravity in the direCtion' M re 
be path, as is well known, will be the parabold’"M’M’, 
point m, if x be affumed to exprefs the | — Mme m', 
the velocity at that point, '‘m, w 
£ 
will be expreffed by Wb +a3 
4 being fome conitant quantity’ previdufly determined, and 
in this inflance depending on the fo 
mare of Ae aaa - Now, if 
- to vided into an infinite number of fmall portions or 
ee pe tet each of thefe be multiplied by the ex 
S oelin which denotes the veloci ity, and whic h 
will have a a aa value ‘for every element into which it is 
. . sant we then it is cuit that the of all 
(or what in the language of analyfief is 
mall Sy 
called cece integral will be tele than if the fame- cea 
d 
co 
ea 
