1820.] Mathematical Principles of Chemical Philosophy. 355 



ingin the direction A E becomes greater than that in the direction 

 B F ; A and E must, therefore, approach to, and B and F recede 

 from, each other ; the system can, therefore, become quiescent 

 only when A and E have attained mutual contact, when A E, 

 E F, A F, are mutually equal, or form an equilateral triangle, in 

 which state the equilibrium of the system is permanent ; for if 

 any particle be disturbed, it returns to its place when at liberty. 



Prop. 10. — The particles of solids being preserved in mutual 

 contact by the force of cohesion, increase of temperature causes 

 expansion, and diminution of heat, contraction. 



Let A, B, C, fig. 6, be three particles of equal diameter, and 

 of the same matter, placed in the same right line ABC. Since 

 in solid matter the centripetal force exceeds the repulsive force 

 of heat, these will be preserved in mutual contact in the points 

 a, b, by the force of cohesion by prop. 1. Let D be another 

 such particle, which touches B in the point C. By prop. 8, 

 cor. 2, the forces with which D is acted upon by A and C will 

 be centripetal. With centre B and radius B A = B C, describe 

 the semicircle ADC; at the point D draw the tangent x y, 

 and join A D, B D, C D. In A D take F D proportional to the 

 excess of the centripetal above the repulsive force (i. e. to C F, 

 prop. 8) ; at the distance A D, resolve this into F E, ED, the 

 direction of the tangent, and the perpendicular to that direction. 

 Take H D proportional to the effective centripetal force (prop. 8) 

 between D and C, at the distance C D, and resolve this into 

 H G, G D; of these forces, only the resolved parts ED, G D, 

 have any tendency to put the particle D into motion ; and since 

 they are both centripetal, and act in opposite directions, D is 

 stationary when E D = G D. Let the temperature be increased, 

 the centripetal force H D is diminished more than F D by 

 prop. 8, .-. D G is less than D E ; consequently D will move 

 towards x (the locus of its centre being always the circumfer- 

 ence ADC;) and since parallelograms upon equal bases and 

 between the same parallels are equal, by completing the upper 

 row of particles, the portion of space occupied by the system, or 

 its volume, will be enlarged. If the temperature be diminished, 

 by prop. 8, the force D H being always centripetal will be more 

 increased than F D ; consequently D will move towards C till 

 a state of equilibrium is attained, and contraction will result. 



Or thus : A, B, C, D, fig. 7, being four equal and similar par- 

 ticles of a solid, since the effective forces which they mutually 

 exert upon each other are entirely centripetal, and perfect con- 

 tact is preserved by the force of cohesion, prop. 1, they will be 

 in a state of rest, when the centripetal forces of the opposite par- 

 ticles, A and C, B and D, multiplied respectively into the ratio 

 of their simultaneous motions, are equal to each other. Suppose 

 them to be in this state of quiescence, by the mutual equilibrium 

 of these quantities, and the temperature, and consequently the 

 density of the calorific atmospheres of these particles to be 

 increabed. By prop. 8, the centripetal force mutually exerted 



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