FORCE. 
_ 
formed on any other line or curve paffing 
had been per ° y ac eGe 
MM’. Let us exam 
by this fuppofition, ner obtain a lefs product than by 
the a one, e divide the line into the fame num- 
ber mall elements as the curve, it is evident that each 
of thele ‘will be imaller, and fo far our objeét being to ob- 
tain a minimum, the advantage is in favour of the ftraight 
we exatnine the multiplier /h +x, we fhall 
hat they increafe ate in ica as ‘they Qcicead, and 
ie we confider the figure, we may obferve that in the ftraight 
line half the elements are quale ste by greater values of 
JSb+ x. Now this effe&t is obviated by the curve, as the 
greater number of its elements are placed in the upper part 
of the figure, w there the multipliers Ae Z nad are compara- 
aed but of {mall value. Were the c e however to be 
made too convex, as the dotted line i in the f gure, this ad- 
ae might be deftroyed by the increafed value of the 
elements, which might more than counterbalance the effect 
of the Suey values of \/ x. It will not be dif- 
ficult now t mprehend thee exiftence o e curve, in 
which the fam. of the factors thus determined fhall be lefs 
In the prefent c a = is found to be a 
parabola, and in pase inftance i e fhewn, that the 
curve anfwerin the above uo mi nimum, is 
moving in {pace determined ; ~~ it is of very 
shieils of inveftigation. The reader will find fo 
of thefe folutioas under the weg of oad Fuxcrion. 
See likesvife IsopERIMETRICA 
The aes explanation a only to the cafe of a fingle 
body, nor did Maupertuis, the inventor of this principle, 
extend it anhen Euler eftablifhes the generality of this 
" principle in his treatife on the Ifoperimetrical aay ae 
fhews that in all trajeCtories defcribed by the action of ce 
inde al forces, the integral of the velocity, multiplied by ie 
element of the curve, ts ans a maximum or a minimum. 
‘Grange firft 
a“ing on each ne and ce mon 
the produéts of the maffes by the ae of the velocities, 
multiplied by ee elements of a {paces deferibed, is always 
either 2 maximum or a minimum. 
Thefe are the ae and le aise principles of the doétrine 
sh have been at different times cxcogitated by 
mott of them 
me 
— 
oe 
t. Tho ae Poke a ring i ee ne i or at leaft for : a 
fhort interval of time, impart u iform motion to a particle 
on which they aé&, provided it - not folicited by. orf ia 
or - and i he fame time free to move in any dire 
. Forces which a& conftantly, and’ whofe catenkty 
remains the fame: a material particle, free to obey the 
a€tion of fuch a a defcribes its path with a motion 
uniformly accelerat 
Forces waste ssn ad are ere seat ha bat 
according to fome known law. ced b 
great meafure foreign to the preen: 
fubject of iovetigation 
Cafe 1. When the force aie a: a“ fame p 
ih feveral forces, as P, Q,S, &c. ( fix. 3. be whole direc- 
tions are M P,M QO, MS, and ttle’ Ma, cy are 
exerted on any point M, the refulting fone a eafily 
aed. by combining the forces by parallelograms two 
and two, till a final refulting force is obtained, which in 
this eae as appears by the figure, is in the direction M R, 
and whofe magnitude is the dae M d. This procefs may 
be fimplitied by taking -) equal and parallel to any 
orce P, ad equal and saute to any force Q, bc to a force 
» &c. then Mc fhall be the refulting force. If the forces 
terminate in M, after pei rae round in a polygon, it 
would indicate that the po was in equilibrium; and 
it is s evident that if for the ulin force we fubftitute its 
oie direftion, the point M will likewife be 
a agp 
mee 
-_5 
6 
= 
So 
* 
Q 
o 
ao 
a 
~ 
oO 
a 
= 
3 
rs) 
cr 
® 
A. 
. 
oO 
= 
_o 
a 
“> 
a) 
“ 
BS 
° 
ba 
a 
‘ot 
° 
G 
i?) 
tS 
= 
3 
de 
by what mode they may mott commodioufly be antre duced 
into analytic calculation. 
Let the angle PM 5 (Ag. > as ‘s QMR=: 
fin, 6 
= a, then : 
: RS in, e: fin, a 
QO: R:: fin, 6: fin. a, 
P Q R 
Therefore it = Boa = a 
n ree forces are in equilibrio, when each is pro 
nl the ratio, and not on m 
tude of the producing forces, it follows that if {everal forces 
are in eal ibrium, au y are sad to vary proportionally, 
they will ftill remain in equilibriu 
If Ps ex $9; andR= 
= 8. 4 
If the dion of Me forces ite yan are at right anette 
Then R* = P? +0 
And P ee ee: 
aa Soe 9 
Q =. tang 6. 
Thefe equations are often of are utility in transforming 
forces into others that are reCtangu “or to refolve any 
force R into two sae which pe ‘ be reCtangular, it is fufh- 
cient to obferv at in. 9, or that 
each credence! force i is the produ of the refulting force R, 
by the cofine of the angle which it makes with this pro- 
ducing force. 
Hitherto we have fuppofed the forces to a& in one plane. 
Suppofe now three forces P, » acting on the point M 
in directions not in the fame ‘plane, and firft let thefe direc- 
ay be re€tangular, thatis, let P and S be at right angles, 
and Q perpendicular to their plane. The producing forces 
Er and 8, he refulting force T’, determined by the a aa 
;s= in are where 4 is the angle formed 
by T ac P. In the fame manner Q and T have their refult- 
« 
