FORCE. 
By ‘equation oe == Pe 
r 
z =i! 
multiply by dx, d ty refpeively, aad add : ud then fince 
xdx+yd pana 
= ai + dy Oy — Pdr. 
But as the andi oe of the force P is fuppofed to vary 
by nl ot Par a as. vig cecil of the diflance r, the in- 
of Pd. 
fe f. way = ot C, then - 
ar raae =a (9+ C) = v* 
This equation contains the principle of the vis viva; 
it is in fubitance he 40th Prop. Se&. VIII. of 
the fame a8 the 
vton’s Principia, from which it is inferred, that. if two 
ave at any one diftance the fame velocity, when 
acted on by any centripetal force, ee ay always have 
the fame. velocity at any other equal difta 
the transformation of the co- aie we obtain 
thefe yung . 
,drPt+rdv=—2(¢4+C) d 
and by eliminating dty ne differential Squaton of the tra- 
jectory becom 
cdr 
ryqo-?r (9+C)—2 
the integration of which quantity gives the curve required ; 
but this can only beeffe€ted in particular cafes. 
This e celebrated inverfe probes of centripetal 
forces, and i is the 42d Prop. of the 1f ea of Newton’s 
du=x 
Principia. A geometrical folution, with many illuftrations, 
is given by Dr. Robifon, in the article ee Sup. En- 
yclop. Britan 
Of Forces oli can adi on a body conftrained to move on a given 
fur face.—Let terial particle defcend from the point B, 
(fg. 19.) and deferibe the curvilinear {pace BM, aravag at 
M atthe end of the time ¢: let ed be the b borin A J 
the vertical axis; fo that ‘AP =x, P 
let BC =. The ey v, in he direct of the oe: 
LS 
ment of the curve, is v = ——, and the augmentation by 
dit 
the force of gravity is dv én the time dt; but this force 
tends to communirate the velonity gdiin the element o 
cae Raps servi be decom- 
e normal, and 
celle of. a curv 3 
-d a a y 
» fince —* is the cofine of the angle: formed - the 
ime TM with the axis of. y- Therefore the aug- 
mentation of velocity i isdu =gdt. i. 
= 28 (y + C). Now at 
t,. where the velocity is 
fuch asis due to fome given height 4, accordin e€ par- 
ticle had or had not fome initial oan pe as either 
k£orh—k 
hencevd v = 
gdy, and by integration v? 
4 
a 
er 
= 
» =oeorv’? =2gh, wheny = é; hence C= — 
Therefore v* = 2 a or v? era (y+ a. 
In the firft cafe gx aye a gravitating 
particle has the fame dace ae the tan- 
city in 
" gent at any part of the curve, as fi it had fallen tau from 
- fame height, and that whatever be the nature of the 
ae en the above equations we obtain 
“= J pso-at wide = | Sago—byan 
~/ (280-8 
From the equation of the ve ds will be known in 
ae y and dy, and ¢ will aie he obtained by integra- 
di 
ne The cafe : the fimple pendulum per be taken as an ealy 
example of the application of this method; a more com- 
plete inveltigation is be r ciemea “for its proper plate 
under the article PEnp 
vi 
dh 
= ce = = 03 being the conftant farce eftimated in the 
General theory of the PindeluieThe egnacons are 
direétion of the tangent of the curve ; s the arc defcribed 
ds 
at the end of the time ¢; and »v = Tt = the velocity ac- 
quired. Let g be the force of gravity ec on ha ma- 
aia point M, fufpended at the extremity of th 
uppofed without weight and inflexible, C vs pain af 
{ulpenfion ‘== « the vertical abfc iff PM = 1¢ 
ordinate of the centre- of ofcillation, AC = a the iength 
of the pendulum, A D = 4 the verfed fine of. the arc BA PY 
(fig. 15.)3 Bthe point of departure, s the arc , andthe 
velocity acquired in the point M rf the end of the — tee 
The velocity v, due to the height D P = 4 
s 
v= Fp 2g b=5) x)’ 
but the arc decreafing as the time ¢ increefes, 
d= 
V2 = _ x) ; 
but ds muft be exprefled in terms of dx, in order to effect 
the neceffary integration. The a erigt of the element of the 
arc being exprefled by ds? = dx? + dy? 
reds Ses (HS, 
the equation to se circle being 
e = yy?” 
—2ax=0, 
ads d 
.andds= =e ’ 
J 24x —~ x? 
— adx 
we obtain dt = 
Vag (bx 1a — x)" 
*he determination is the time “therefore d depends on the 
integratio onof the above exprefli on, which we fhall not en+ 
ter into at prefent. This — effe&ted, gives 
ead @ ° AS (3 y+ (3 es 
. 4 arc (co. = 2 ="). 
At the limit x = 4, likewife ¢ = 9, and when x = e 
a — 
arc (cos. — a an = © Or 3%, 5 v Kc.3 © being 
the femi-circumference w 
ad. = uting T 
for the value of ¢ refulting fon the pene ee 
