OF FUtSSURE AND EQUILIBRIUM. 



37 



is equal to that of B in describing BF (270, 28J. And in 

 the same manner if the common centre be transferred from 

 G to H by the motion of C in CI, and then to K by that of 

 D in DL, K will still be the place to which the single body 

 ■would be removed by equal momenta successively commu- 

 nicated to it. • If the motions of the separate bodies be not 

 successive but simultaneous, K will nevertheless be their 

 common centre of inertia; and if the motions of the single 

 body be communicated to it at the same instant, their- 

 joint result will still transfer it to K, since AK is the result 

 of the motions AG, GH, HK (226). Therefore the motion 

 of the single body always coincides with that of the com- 

 mon centre of inertia of the system. 



275. Theorem. The centre of inertia of 

 a system of bodies moving without disturb- 

 ance is either at rest, or moving equably and 

 rectihnearly. 



For the result of any number of equable and rectilinear 

 motions being also an equable rectilinear motion, as may 

 easily be shown, by combining them in pairs, from the 

 properties of similar triangles, the motion of the centre of 

 inertia will also be equable and rectilinear (274). 



276. Theorem. If parallel lines be drawn 

 from each of a system of bodies, and from 

 their common c^ptre of inertia, to a given 

 plane, the sum orthe products of all the bo- 

 dies into the segments of their respective 

 lines, will be equal to the product of the sum 

 of all the bodies into the line drawn through 

 the centre of inertia. 



Suppose each body to describe its segment in the same 

 time, then when they arrive at the plane, their centre of 

 inertia will also be in the plane, and the product of each 

 body into its segment will represent its momentum ; and 

 . .the product of their sum into the distance described by the 

 centre of inertia will be the momentum of .a single body 

 equal to their sum, and coinciding always with that centre ; 

 but these momenta have been shown to be equal (274). 

 The theorem may also be more directly demonstrated. 



277. Theorem. The distance of the cen- 

 tre of inertia of any triangle from the vertex 

 is two thirds of the line that bisects the base. 



The triangle being supposed to be divided by lines pa- 

 rallel to the base into evanescent portions, it is obvious that 

 the centre of gravity must be in the line which bisects them 

 all ; and the sum of the products of each portion of the 



line into the corresponding element, divided by the whole 

 area, will be the distance required ; but the whole area be- 

 ing i,a:cx, the area of each portion is axx', and the product 

 ax\T', the fluxion ax'ir, and the fluent i«,r' ; which, divided 

 by the area, gives ix for the distance required. 



278. Theorem. The place of the centre 

 of inertia of three or more bodies is not af- 

 fected by any reciprocal action among them. 



For since, in all reciprocal actions between two bodies, 

 equal momenta are communicated in opposite directions 

 (269, 270), the joint effect of each pair on a single body 

 supposed to be placed in the centre of inertia of the system, 

 will be to destroy each other, therefore its place, and that of 

 the centre of inertia (274), will be the same as if no re- 

 ciprocal action existed. 



279. Theorem. When bodies of the 

 same kind attract or repel each other, the 

 force is in the compound ratio of their bulks. 



For each particle of A, being actuated by each particle 

 of B with a force equal to unity, is actuated by the whole of 

 B with a force equal to B, and the whole of A with a force 

 equal to A.B. 



280. Theorem. If two bodies act on 

 each other with forces proportional to any 

 power of their distance, the forces will also 

 be proportional to the same power of either 

 of their distances from their common centre 

 of inertia ; hence the reciprocal forces of two 

 bodies maybe considered as tending to their 

 common centre of inertia as a fixed point. 



For X, the distance of either body from the common cen- 

 tre of inertia, being in a constant ratio to the whole distance 



y, may be called ay, and the force 2/"= ( - I "=;x" : a", 



which is to x" in the constant ratio of 1 to a". 



Scholium. It is observed that all known forces are re- 

 ciprocal. This circumstance is generally expressed by the 

 third law of motion, that action and reaction are equah; 

 but it often happens that the difTerence of the magnitudes of 

 the two bodies being very great, the motion of the greater 

 may be disregarded. 



SECT. VII. OF PRESSURE AND EQIIILIBRIUM. 



281. Definition. A pressure is a force 



