FOR 
laa red as an ry ae but which, though very obvious, 
mained ee unobferv 
i hi 
s: if feveral — have a tendenc cy to motion, 
5 aaa this theorem -/ an ea oat ; fuppofe 
two vinelattic bo a or balls, whofe celles e M and m 
meet in the fame direction with sangria V.0 Vy ieee 
required tie common velocity after t ck. 
Decompofe the velocity V into x a V—-xs, 
and the vanes v intox and v — x. 
The velocities V — x will a deftroyed, but 
as they alone would ave masiveained the bodies in equilibrio, 
they muft be inverfely as their maffes ; hence but one in this 
cle mult be taken with the contrary fine: hence 
If a body or material point M (PlateXXVIIT. eae 
Jig. 6.) befu ippof2zto move from She ay the line AB’ byt 
Sve velo 
ra 
~ 
~ 
,and expre ff. 
elementary {pac = OG, may | 
x 
ed by ae x bemy th the {pace at the ti 
of a new inftant « ¢, (in which is delcribed the oda an): pee 
increafe of force or velocity which 
dt; P reprefenting the ined, of the fe 
therefore at the commencement of this new 
+ +] 
prefied by 
The velocity, t 
inftant, will be = + P.d#; but this initial velocity may, 
¢ 
by icine s theorem, be decompofed into two others, of 
which one will remain and the other be deftroved ; and the 
velocity deftro ed will be fuch, that if M had een ag ose 
by t 
that alone, it would have remained’ in ay la 
the velocity remaining is evidently — +d, ora and theve- 
locity deftroyed therefore P.dt — d. - 3 fince thefe 
two quantities together equal P. d# + a 
If the point M therefore had been folicited only by the 
® Sane a 
force P. dt ~— d. —, it would haveremainedinequilibrium ; 
dt 
hence 
P.di—d. oe r) 
dt 
ae ek 
which is the general differential expreffion for the acce- 
fe 
lerating force. 
Since two forces P. dt, andd. - » ating on the point 
M, keep it in equilibrium, then if the point be fuppofed to 
vary its pofition an ey = eae. ous which may 
be here taken equa e elementary 
theorem of La Place, eh ie meltiptied into the ele- 
' Pidx. 
Cc E. 7 
ment dx fhould equal zero; hence d « = fad! ——~- 
. P. ast = 0. 
And by integration 5 * = C +, twice the integral of 
dx . : 
But TF, = is the fquare of the velocity. There. 
fore the fquare of the velocity is always equal to twice the 
wale of the ecciaa force idee into the slement 
Ta ie obferve that the theorem 
of a’ Alembert | is ae of itfelf {ufficient to folve a problem, 
fince it is always neceffary to derive fome condition relating 
to the equilibriwm from other confiderations e diff- 
culty of determining the forces, and the law of the ae 
brium o efe forces, renders this Bip lieadce fometim 
more diffic ult, and the procefs more tedious, than if the foli- 
tion were i rmed by fome prenciple Jefs fimple and direét. 
It was by conihiane the above principle of d’Alembert 
with nies of virtual velocity, t that La Gran ge was enabled ta 
pees the general equations reg the 
ona 7 ftem of bedies. 
ce met 
tot orces which 
He thus defcribes the nature of 
o form ee ale of the manner in which 
dee principle are a e fhould recoll:& that the 
general principle of te a aise 
when a f th 
on oO and by that of the moving forces b 
w which ney are ioliied: Sus there will be au equilibrium 
between thiefe forces, and t fee or refiftances which 
refult from the motions wich oft by the bodies from 
one inftant to another. Hence it follows, that to extend to 
the motion of a fyftem of bodies, the formule of its equi- 
librium, it is ae to add the terms due to thefe laft- 
mentioned force 
I snfider the velocities which every particle 
as in the diredtion of three fixed re€tangular co- ordinates, 
the decrement of thefe velocities will reprefent the motions 
loft in thefe dircGtions, and their increments wi the mo- 
tions loft in the oppoftte directions herefore the refulting 
preffures or forces of thefe motions deitroy lbe ex- 
n 
the Fi the peta sre from t 
formula thus obtaine 
