OF PRESSURE AVD EQUIMBRIUM. 



39 



This case is somewhat intricate, and may be thus demon- 

 strated. Draw BF parallel to CD, and GHI to AE produ- 

 ced to F, thenHE:KE::BE:FE (121), andDE:DL:: 

 KE : HL :: FE : IL, therefore HE : HL :: BE : IL (l?), 

 and BI is parallel to ED. Now A : B .: BC : AC r: FK : 

 AK : : IH : GH, and A.GH=B.HI. But by what has 

 been already demonstrated, the pressure of A and B in the 

 directions AE, BE, are A.GH and B.HB, DH representing 

 the force of gravity, since the lines are parallel to the forces 

 exerted ; and A.GHi^B.HI : therefore the forces of A and 

 B at E being B.HI and B.BH, their result will be parallel 

 to BI, or in the direction ED, and will therefore be wholly 

 counteracted by the rod DE, without any tendency to turn 

 it round D. 



There is another simple and elegant mode of demon- 

 strating the property of the lever, which deserves to be no- 

 ticed. Supposing the arms to be a little bent, and the forces 

 to act perpendicularly to them, so that their directions may 

 meet in a distant point ; then if their actions be imagined 

 to be concentrated in that point, it will be easy to show 

 that in order that the resulting force may pass through the 

 point of suspension, and that an equilibrium may be thus 

 produced, the forces must be inversely as the perpendicu- 

 lars falling from that point on their directions ; that is, as 

 the arms of the lever inversely; and this will be true whe- 

 ther the lever be more or less bent, and consequently even 

 if it be not bent at all. It is not however strictly shown in 

 this demonstration, that the effect of the forces must be 

 the same as if (hey were applied in the point where their 

 directions meet, and a link appears to be still wanting in 

 the chain. 



286. Theorem. A system of any num- 

 ber of gravitating bodies, or a mass composed 

 of such bodies, will remain in equilibrium 

 when its centre of inertia is in tiie vertical 

 line, passing through the point of suspension. 



Let us first suppose the number 

 of bodies to be three ; let A and 

 ■g ^ ^^ 5° connected as to remain in 

 equilibrium on their centre of 

 ^ inertia C ; and let this centre and 

 the third body E be in any way 

 connected with the point of suspension D : then since C 

 supports the weight of A and B, it will retain E in equili- 

 brium whenever the common centre of inertia F is in the 

 vertical line. And the same may be demonstrated if the 

 bodies be connected in any other manner : for instance, if 

 alt the bodies be suspended from D, and retained in their 



places by the lines AB, AE, BE. Then A will counter- 

 poise a body at E of which the weight is to its own as AG 

 to GE (78), or HF to FE, and B a weight in the propor- 



A HPJ-R IF" 



tion of IF to FE, and both, a weight:::—^ ^ — — 



FE 



A.(CF— HC) + B.(CF-|-CI) _ (A+B).CF B.CI — A .HC _ 

 FE ~ FE + FE" ' 



but A : B :: CB : CA :: CI : MC, and B.C1=A.HC; there- 

 fore the last term vanishes, and A and B support a weight at 



CF 

 E equal to (A-f-B).— ,, or equal to E : and the effect is the 



same as if they were united in C. Therefore either of the 

 bodies may be divided into two, and the equilibrium will 

 remain, provided their centre of gravity be in the place of 

 the single body : and thus the number of the bodies may 

 be increased without limit. 



The proposition may also be more generally and com- 

 pendiously demonstrated from other properties of the centre 

 of inertia. Imagine the fulcrum itself to be suspended by 

 a veilical thread, and let the centre of inertia of the system 

 of bodies be so placed, as to be in the same right line with 

 this thread; there will then be a perfect equilibrium: for 

 the motion of each of the bodies in consequence of the 

 action of gravitation, and of course the motion of their 

 common centre of inertia, would, if they were wholly at 

 liberty, be in vertical lines; and since the mutual con- 

 nexion of the bodies suspended, causes only a reciprocat 

 action between them, it can have no effect on the motioiv 

 of their common centre of inertia: consequently the thread) 

 acting in a vertical line directed to that centre, will render 

 its descent impossible, and completely counteract the whole 

 force of gravitation, so that no force will remain to produce 

 any other motion. Now since the fulcrum suspended by a 

 thread would remain at rest, it is obvious that it may be 

 fixed in any other manner, and the equilibrium of the 

 system will remain undisturbed, as long as the centre of 

 inertia is in the same vertical line. 



Scholium. Hence the place of the centre of inertia of 

 any body may be practically found by determining the 

 intersection of any two positions of the vertical line. 



287. Definition. Tlie centre of inertia 

 is also called, on account of these properties, 

 the centre of gravity. 



288. Theorem. If a sphere or cylinder 

 be placed on another, the equihbriuni will 

 be either stable or tottering, accordingly as 

 the height of the centre of gravity above the 



