1823.] Mathematical Principles of Chemical Philosophy. 253 



H G, the elastic force of the atmosphere of F is equal to that of 

 G in the point H. Bisect D F in K, and through K draw L M 

 at right angles to it, which will cut D I, D H, in L and M ; join 

 L F, F M ; then, since D K, K F, are equal, the atmospheres of 

 D and F are equally elastic at that point ; because D L is equal 

 to F L, they are equally dense at L ; also at M ; or if any other 

 point n be taken in L M, and D u, F n be joined, the forces will 

 be equal at that point ; and if the solid figure be taken, the 

 elastic forces of the atmospheres of D and F will be equal in 

 every point of the hexagonal plane passing through K, to which 

 D F is perpendicular; therefore, the entire repulsion between 

 those particles takes place in that plane ; and the same may be 

 proved of ail the other particles ; therefore, since the angles 

 L F K, K F M, are constant, as the density varies, all the lines 

 L F, F K, L D, &c. have to each other a constant ratio ; and by 

 Sect. 3, Lemma, density at K : that at M in a constant ratio, 

 and since the elastic force at K is inversely as K F, that at L is 

 inversely as K F, and the same may be proved of the entire 

 repulsive force ; therefore, the entire repulsive force between the 

 two particles D and F will be inversely asDF; therefore, by 

 Newton's Princip. lib. 2, Prop. 23, the elastic force of the gas 

 will be inversely as the volume. Q. E. D. 



Cor. 1. — If the compression be such that the force of attrac- 

 tion between the particles produce any sensible effect, having 

 given the ratio of the forces at any given distance, the force will 



be as = — ■=£> each term being multiplied by the force at that 



given distance. 



Cor. 2. — If the gas be highly compressed, its elastic force will 

 increase in a higher inverse ratio than that of its volume. 



Scholium. 



If the force of attraction vary inversely as the cube of the dist- 

 ance, the cube of the compressing force will be as the fourth 

 power of the density. If the force of attraction be as the fourth 

 power of the distance inversely, the cube of the compressing 

 force will be as the fifth power of the density. 



Prop. IV. 



Gases will be expanded by heat, and contracted by cold, their 

 elastic force being constant. 



Let the elastic force of a confined portion of a gas be balanced 

 by the pressure of the air ; heat it, and the elastic force of the 

 calorific atmospheres of its particles will be increased; therefore, 

 the particles will have a tendency to separate from each other; 

 the gas will expand ; but the force of each particle is inversely 

 as the distance ; therefore, a distance will be found, such, that 

 the gas will again be in equilibrio with the external air, and con- 



