FORCE. 
learn. the aaa of the motion, ‘and determine the trajeCtory 
which the body defcribes, when the intenfity and direction 
of the forces are given. Curvilinear motion is thus reduce 
to two or three rectilinear motions, according as the curve 
is of the -— or double curvature 
the traje€tory lies in the fame plane, the motion 
e(t)isy=f (2): 
prefernted by-three equations, x = ¢ 
v Ch eee by ads t, two equations will be Send 
in ter Hy Py z, which will be thofe of double 
Lae tue defaribed i the ioe. The ne i) 
‘), gy =f z= : (¢)- 
; ap are: which aé&t on ie ma- 
terial particle M, - t 
B's B"s y's y"’3 be the sain formed by their dire&tions 4 
the inca axes Of x, y, Let each force be refolved 
into three others parallel to thefe sae axes, namely, 
‘X,Y, Z, then each of thefe forces will imprefs on the 
ody an elementary impulfion, each in its own direction. 
Hente thefe equations; 
X = P’ cos. «a! + P" cos. a" + &e. 
“Y = P' cos. B! + P! cos. B! + &c. 
Z-= P'cos. + + P" cos. y! + &c. 
At the end of the time #, the point M, in that point of 
its trajeCtory which has for its a Hy Vy By Wi 
are fo that in this 
poi e may conceive the body in repofe, and receiving in 
the cretion of the axis x an impulfion producing the 
have in the dire€tion of x a velocity = 
velocity —— ; this velocity fhould increafe during the time 
7 
dt, by a quantity = =a (= ; by the effet of the con- 
ftant forces P, P’, &c. fo as to become = = 2 +d 
=) . But,-as we have already feen, the nies com- 
d x 
municated in the diedion of the axis x is equal ae 
+ X dt; and as the effects of oo other tale 
forces ‘are independent, the forces X, Z, 
ey thig. Hence. the velocities X d t,-.d are 
a - 
ual. Reafoning in the fame manner as to the other forces, 
we Chie the following equations. 
Kapa a(S) 
(2) 
Ydi=d\-- (A!). 
| pane(24) 7 
of thetime#. Let o', as 
Or taking d¢ conitant, 
which are the general formula of motion for a body 
paises in free 
equations are fafficient to inveftigate all 
ia tace: of the motio 
the cir- 
of x and y, "S ae Y beiee ea either conftant or vari- 
able, we have then ie to eliminate ¢, the time from the 
d*y 
»Y =e This being effected, 
and the integrations a an equation is obtained between 
x and y; fimilar relations may be obtained between x and 
ay 
— 
two equations X = 
t, and y and ¢, then 
di? : 
the directions of the axes of x and y; whence the real 
velocity may be concluded, which will be v = > 
J (A+ Q'}. 
A pees more ufeful on many occafions, may be thus 
obtained 
Tn the equations 
2 
will pive the velocities in 
— 
— 
a a a 
ae es 0 dF 
Multiply by d x, dy, d x, as follow 
Xdx+Y¥dy+ cde a ete t aye bind dla 
but 
is the differential of 4 
the numerator of the fecond member of — equation 
(dx? + dy? + dz*), 0 of 3 é 
pe by taking the integrais, we have —~ = »° 
aF Pr 
+2/(Xdx +Y¥dy+Zdz). This ieee whit 
is fimilar to that we have already ufed in the of 
Lr ri cannot give an ex 
z be ane 
— Ieneuage fee hints, except i 
un 
3 or, to ufe 
ts exa&t fluent can be 
«AE, aie efore, X, Y, Z, be a of x, ¥, %, the 
aay conditions Toul fub 
aX dY dX d Z aZ. || 
—_ = ss, . = ——, and 
dy du ds dx dz y 
v + C. : : 
on conftant quantity . ica on the initial velocity, 
or on the velocity a en ] 
equation comprehends the prt pk of the ‘ confervatio 
virium vivarum.’ 
his ee is in fa&t the fame as the goth prop 
of the firft book of Newton’s Principia. The. reader will 
find a very oie geometrical demonftration, sage ag 
with many va aluable remarks on thefe theorems, in 
Robinfon’s Mechanical Philofophy, and in the Supplement 
to the Encyclop. Britan. 
One of the moft fimple examples of the eat of 
