FORCE. 
B?2 
2° a 
vaav (4) {6+ Gast 
ay 
(34 
113-5 pa te. 
e fa 3) 3 + &e. 
When -s are of ofcillation is. very jane 7 may be 
ng a the whole ofcillation 2 'T’ beco T 
7: -. See Penputum; and for the.curve of quickeft 
deloent, fee Cycom and dualytic Function. For the 
motion on an inclined plane, eae is a more fimple cafe of 
the fame kind; fee TNGEEXED 
In the above examples the eal ‘has been fuppofed to 
arife from the aétion of gravity alone, but the 7 may 
be anaes = extended to any forces whatever 
A bo 
d et be conftrained | move in a curve ec rface 
ec 
° 
0 
Pty 
3 
° 
ba 
3 
a 
a the ail of the 
= y, and N equal to i nor a force = AP = ‘the re- 
aétion of the curve : i : oF are the cofines of the angles 
which the normal makes with the axes « and y, therefore 
the producing a equivalent to N in the direction of 
dx 
thefe axes are — N- = J and N Te the firftis negative, be- 
ds 
caufe it tends to diminifh the value of x. The body 
therefore be confidered as free and folicited by the eek 
X—N —= “J rey and Y + ni » and the equations of varied 
ds 
motion care given become 
a dy d’y d x 7 
Tama Th oJ aven? 
dxy dy dy 
d(, )=(®-NZ)andy a) = (¥ S (m) 
+ x 22) dt 
By combining with thefe the equation of the curve, 
three = = obtained between the four variable quan- 
tities By Jot d N, fo that by — and elimination 
ed between _any tw 
them. 
ode "art of the above 
ation 
terms containing N will ca oer and an eens 
fimilar to ks already found, will be claret 
~+Y¥dy) 0 
wv + 29; 
which contains oie a es of the vis viva as was ob- 
ferved above MecuHanies 
The value of C depends on the values of vw and ? a 
fome given inftant ; let v! and ¢" be thefe values at that in- 
,Azv? ~— 29, 
fince ov? = v 4 ; 
(,) don the co-ordi- 
nates of the extreme points, fo that the a at ¢he fe- 
> 
pak) 
3 
s 
rte 
as 
o 
i] 
cS} 
BS 
an 
3 
09 
e” 
ll 
qs 
“ws 
cond inftant is given by the apes! . the firft, and by the 
espa a oo ie ein uu hi the velo-° 
ae re of the c ve deferibed, 
i y been conftrained to move in an 
through thefe points, the veloc 
point will be the fame. 
If the body i is not etolicned by the continual aGion of any 
force, but arifes ee from fome aie i eee its ve+ 
locity will be conftant, for when »,C=v% 
} = of gravity 8 eae in rea ia for if X =a, 
= aa if B be the point of de. 
) then making Y= B 
v= 0 2g and w= 2 ¢ 
x MI. The ee as found above. 
The preffure which the body exerts on the curve is equal - 
and oppofite to the force N ; to determine it let d¢ be taken 
as conftant, and dx as variable, then the preceding equa- 
tions (m) become 
eatin oon a hate of ref A 
=z & as befo = 
(yk) = 
ae sz ad 
= ae d i= xX foment N . ae 
dt.dy—dy.ad't 
tee = ¥4NG%. 
To pei raed d’ ts sees Pa frit DA d ae the fecond 
by dx, and fubtra&, then fince 
dxda@y - dz? +. “ly 
a Yéa— Xap 4N( AP) vas 
_ N ds. The value of N aon be determined in 
erms of X and Y, and of the differentials depending on the 
wae of the curve, but the formula may be rendered more 
fimple, by introducing the radius of curvature R, which is 
always equal to By fubftituting therefore for 
dxd'y 
dxd’y its value = we obtain this equation ; 
v Xx St ee — VY dx 
ee (2) 
When, therefore, } {fh eon 
dtom 
a curve furface, whofe equation y= OK is given, we mak 
to the e general eq 
ds° 
sa mets (det Ydy + 2am 
and for X, Y, y and dy, re paige ie values in terms of 
x3 and deduce fie value of v 7 eae by a 
fimilar fubftitution in the Pp: mecedlite on (n), we fhall 
obtain by integration the eee in the eee of every 
~— at se end of a ie ia ti 
s b that the preflure — N, 
ac ae dy eee pees os the curve it defcribes, is coms 
ofed of two parts; one en depending on the velocity 
of the body, the other on the aie forces by which 
it is folicited; the other is the fum of the ferees — xe ZY 
q. 
Y rae which eauibined produce the accelerating forces in 
the direétion of the normal. If the body is not fubje@ to 
the action of a any ae forée, its eae can onl 
arife from an original impulfion, and will therefore be uni- 
form; the — Dee fore -will be ex preffed by the firft 
4 
part 
