dP PRESSURE AND EQUrHBRIUAf. 



38 



counteracted by another force, so that no 

 motion is produced. 



282. Definition. Equal and propor- 

 tionate pressures are such as are produced by 

 forces which would generate equal and pro- 

 portionate momenta in equal times, 



283. Theorem. Two contrary pressures 

 will balance each other when the momenta, 

 which the forces would separately produce 

 in contrary directions, are equal; and one 

 pressure will counterbalance two others when 

 it would produce a momentum equal and 

 contrary to the joint momenta which would 

 be produced by the other forces. 



Conceive the forces to act alternately during equal eva- 

 nescent intervals of time, then the one will at each step de- 

 stroy the preceding effect of the other ; let this action be 

 doubled, and the forces will become a continual pressure, 

 the total effects still destroying each other. If this reason- 

 ing be thought unsatisfactory, the proposition may be as- 

 sumed axiomatically, or may be deduced from the equality 

 of the effects of equal causes. 



284. Theorem. If a body remain at rest 

 by means of three pressures, they must be re- 

 lated in magnitude as the sides of a triangle 

 parallel to the directions. 



Suppose the body A to be suspended 

 ,. by the thread AB, on the inclined plane 

 . C AC, to which AD is perpendicular, BD 

 being the direction of gravity, then that 

 the force BD may be destroyed, it must 

 be opposed by an equal force DB, and if 

 DB be composed of forces in the directions 

 DA, AB, the forces must be as those sides, or as the sides 

 of the parallelogram of which DB is the diagonal ; and the 

 same is true of any other pressures. 



285. Theorem. If two gravitating bo- 

 dies be suspended at constant distances from 

 each other and from a given point, they will 

 be at rest when their centre of inertia is m 

 the vertical line passing through the point of 

 suspension : and the equilibrium will be stable 

 when the centre of inertia would ascend 

 in quitting the vertical line, tottering when 



B 



it would descend, and neutral when it can- 

 not quit it. 



Suppose the bodies A and B, of which C is the centre of 

 inertia, to be suspended from D by the threads AD, DB, and 

 to be retained at the distance AB by the rod AB, and let 

 C be in the vertical line DC. Let the force of gravity be 

 represented by DC, then AD will repre- i> 



sent the action of the thread, and AC the 

 pressure exerted by A on any obstacle 

 at C (284) ; arid in the same manner BC 

 will represent the pressure of B in the 

 direction BC, supposing the weights A and B equal ; but 

 since they are unequal, the ratio of their masses must be 

 compounded with that of the forces, and A.AC will repre- 

 sent the actual force of A, and B.BC that of B ; but " 

 A : B=:BC : AC, and A.AC=B.BC; ; therefore the pres- 

 sures are equal, and the bodies will remain in equilibrium. 

 But if the centre of inertia ascended towards either weight, 

 as A, the segment AC, which determines the action of A, 

 would be increased, and BC lessened; therefore the weight 

 of A would prevail, and the centre would return to the 

 vertical line. But supposing C above D, the rod and 

 threads must change places, and the same demonstration 

 will hold good; and since in this case the weights pull 

 against each other, the prevalence of A when the centre of 

 inertia descends towards its place will draw it still further 

 from the vertical line, and the equilibrium will be lost. 

 Now the distance of C J) ^ 



above or below D is ot " q 



no consequence to the A 2 - 



equilibrium ; therefore ^ 



when that distance vanishes, and the thread and rod are 

 united into one inflexible right line or lever, those points will 

 coincide, and there will still be an equilibrium ; which may 

 properly be termed neutral, since no change of the position 

 of the bodies will create a tendency either to return to their 

 places, or to proceed further from them. But the case of 

 an inflexible right line is pe.fectly out of the reach of expe- 

 riment, since the strength necessary for the inflexibility of 

 a mathematical line becomes infinite, and that in an infi- 

 nitely small quantity of matter. If any other mode of con- 

 nexion by inextensibleand incompressible lines be imagin- 

 ed, there will still be an equilibrium ; for instance, if AC, 

 BC, DC, be rods ; 

 and AD,DB, threads; 

 and C the centre of 

 suspension ; or if AE, 

 BE, DE, be rods; 

 and AD, BD, threads. 



