710 



K1CHANICAL PHILOSOPHY- STATICS [OSKTBB or ORAVITT. 



Hence X, -P, cot. a,andY, - I 1 , tin. a,- 



In imtUr 

 manner it may b 

 ahown the force 

 P, acting at A 



' 



may be replaced 

 by the force* X, ( 

 acting at j 



Oin th 



OX and OY, and (. 

 the couple who*e 

 moment i- 



V,. "tore 

 X* - P, co*. a, 

 and Y.. P, sin. a,. 



Likewise P. actinj at A, may be replaced by 

 Y, acting at in tho directions O X and ( ) Y and the 

 couple whose moment is Y 3 r x X 3 i/ 3 , where X, 

 P. cos. n s and Y 3 P s sin. a s . 



Lastly, P 4 acting at A 4 may be replaced by X. and 

 Y 4 acting at O in the directions O X and O Y^and the 

 couple whose moment is Y 4 z 4 X 4 y 4 , where X 4 P 4 

 oo*. a 4 and Y 4 P 4 sin. n 4 . 



On the whole, therefore, the four forces P,, P,, P., 



an, I P 4 , acting at A,, A,, A ft , and A 4 , in the directions 



A, P,, A.,Pj, A 3 P,, and A 4 1' 4 , have been replaced 



by four forces X ,, X . X,, and X 4 , acting at O in the 



whoso re.sultaut is a single force equal to 



X + X. + X s + X., acting at O in the direction 



<>\ ; bv'four forces Y,, Y,, Y,, and Y 4 , acting at O 



in the direction O Y, whoee resultant is a single force 



equal to Y, + Y, + Y s + Y 4 acting at O in the direo- 



t V . !,._.! h-r with four couples, whose resultant 



'.i-, Prop. XVI., will be one whose moment is e^uil 



to 



(Y. r, X, !/,) + (Y r., X y,) 

 + (Y, z s XVy,,) + (Y 4 * 4 - X 4 y t ). 



The two force. (X, + X, + X, + X 4 ) acting at O 

 in the direction OX, and (Y. + Y s + Y 3 + Y 4 ) acting 

 at O in the direction O Y, will have a single resultant, 

 which may be found by Prop. I. 



In order that the body may be in a position of equili- 

 brium when acted on by tho forces P,, P 2 , P s , and P 4 , 

 applied at A,, A,, A s , and A 4 , in the directions A, P t , 

 A.. P,, A 3 P., and A 4 P 4 , this resultant must be zero, 

 and so must the moment of the resultant couple. 



In this case, excluding those conditions which belong 

 to t!. . referring to Prop. IX., we shall have 



two conditions for the forces acting at O in the direc- 

 tion* O X and O Y . 



And Y, 1 + Y" + Y* + V 4 * =o 

 The moment of the resulting couple being also zsro, 

 gives us a third condition. 



(Y, 



(Y, x s X, 



(Y 8 x s - 



= 



, 



sin. 



,r, 4 x 4 - 44 -. 



Where X, - P, COS. n,, X 2 = Pj COS. o.j, X 

 cos. o s , X 4 - P 4 cos. n 4 Yj = P, sin. <',, Y = 

 a t , Y 9 - P s sin. ..and Y 4 - P. sin . 4 . 



The above reasoning may readily be HUOMIM bom 

 four to any iiuinlier of forces. 



\V C . have, for the sake of simplicity, drawn the direc- 

 ,,f (I,,- f,.- . l' ft , and 1' 4 , in such a manin-r 



that their co-onlinates and resolved portions should bo 

 positive. In other cases we must rfinnnoer that if the 

 resolved part of any force act in an opposite direction to 

 we have drawn, it must l>c f ;ve ;* 



and if i..- ..i l^.th of the co-ordinates of tho points of 

 application of the force be negative, we have only to use 

 the negative sign. SuUtituting these signs carefully in 

 the above formula-, we may extend thorn so as to include 

 r praaible ease of any number of forces acting on a 

 )lv in the same place. 



<>K CU \VITY lly 1'rop. XVII. we 

 found that if any number of parallel forces represented 

 in magnitude and direction l,y P,. P., I',, .l-c., P. (Fi^'. 

 C9), acting on a rigid body in the same piano, at point* 

 s*. MM, p. ?w 



A, A*, A , .'. l_to ectangular axes, 



. f.,y.,\>e the rectan- 

 gular co-ordinates of A,, A.., A,, &;c., A. referred to 

 these axes, the resultant lon-e will \>v v<i\\:\\ to P,' + P,' 

 -f. P.' + P.', acting in a diiv<-t:on parallel to 1' . I 1 . 

 <tc., and applied at a point whoso co-ordinates z and y 

 may be found by the formulae. 



P, S. + P. X. + P. X t + &&, + P. X. 



Pi + Pa + P. + Ac., + P. 



P.y. + '.y +P.y. + *"., + p. y. 



- ic., + P. 



From these formula it is obvious that the position of 

 the point of application of the resultant of the parallel 

 forces is independent of their direction. Hence, if 

 of the forces represented in magnitude and direction by 

 A, PI, A, Pj, A, P,, and A, P, be turned through 

 the same angle a, about the points A,, A 5 , A.,, Arc., A,, 

 into the position shown by the dotted lines, the point ol 

 application of the resultant will not move. 



For this reason this point is called the centre of the 

 l><irtilM force*. 



We have seen that the weight of a heavy body is pro- 

 duced by the earth's attraction on each of the material 

 particles of which it is composed, and that this attraction 

 is the same for all kinds of matter. Hence, a cubic 

 inch of iron weighs more than a cubic inch of wood ; not 

 because the earth's attraction is greater for particles 

 of iron than for those of wood, but because a cubic inch 

 of the former contains a greater number of gravitating 

 particles than the latter. 



The attraction which the earth exerts on all masses 

 near its surface, on account of its larger relative mass, 

 is so great, that for all masses which are not very large, 

 we may neglect tho attraction which the particles of 

 these masses or those in their neighbourhood exert on 

 one another. Wo can, on this account, rapidly deter- 

 mine, with considerable accuracy, the direction of the 

 earth's attraction, or yntrily. 



Let a email weight of lead or brass P (Fig. CO), be 

 fixed to one extremity of a thin flexible string A P, and 

 d from a fixed point A. The weight P, 

 A" after oscillating for some little time, will, if 

 undisturbed, rest in such a position that the d 

 A P shall point to tho earth's centre. Such an 

 instrument is called a plummet or a 

 and the lino in which it rests, th If two 



plumb-lines bo suspended from points situated at 

 ciill', lent paits of the earth's surface, they eannot 

 be exactly parallel, but must make a certain angle 

 \vith each other, unless tho one point lie in the 

 antipodes of the other, in which case the direc- 

 tions will bo in tho same straight line. 



For all practical pui account of tho 



_ comparative greatness of the earth's radius, wo 

 Pi may say that two plumb-lines of any ordinary 

 length, suspended within any apartment, however large, 



