354 Mr. Emmett on the [Nov. 



ing D will be the same as if the altitude D E were continued in 

 every direction round it, and the same attains with respect to the 

 other particles ; consequently by the pressure of the column, or 

 superior part of the atmosphere, E M, the atmosphere of the 

 particle D, will have a tendency to dilate in every direction D O, 

 where its altitude is less than 1) E; by this tendency of the atmo- 

 spheres of D and C, they will have a tendency to recede from 

 each other, which, for the same particles, will be proportional to 

 the area A R O P', see fig. 1 to prop. 6, cor. 1. 



Prop. 8. — To find the ratio between the centripetal force and 

 the repulsive force of heat, at different distances from the same 

 particle. 



Let A, fig. 4, represent a particle of matter, ABO the radius 

 produced : describe the curve D C P such that the ordinate C N 

 may always be inversely as A N^, the ordinate will be propor- 

 tional to the centripetal force, at the distance represented by the 

 corresponding abscissa. Describe the curve E F Q such that 

 its ordinate F N shall be proportional to the area A R O P', 

 prop. 6, cor. 1, the ordinates of this will, therefore, be propor- 

 tional at the distances which are proportional to their respective 

 abscissae to the elastic force of the calorific atmosphere of the 

 particle ; and by prop. 6, cor. 1, the curve E F Q approaches the 

 asymptote A O more rapidly than D C P. 



Cor. 1. — Hence the effective force at any distance A N will be 

 as the difference between the ordinates of the curves at that 

 distance. 



Cor. 2. — If E B be less than D B, the force will be centripetal 

 at all distances, and in contact ; and if N move from B towards 

 O, the efficient centripetal force will increase to a certain dist- 

 ance, beyond which it decreases. 



Cor. 3. — If E B be equal to D B, the effective force will be 

 centripetal at all distances, with the variations of cor. 2. 



Cor. 4. — If E B be greater than D B, at a certain distance, 

 the curves will intersect each other, or have equal ordinates, the 

 forces will, therefore, be equal ; between that point and B, the 

 effective force will be repulsive ; beyond it, centripetal. 



Prop. 9. — If the particles of matter, being spherical, were 

 under the influence of a centripetal force only, their arrangement 

 would be such that the straight lines which join their centres 

 form equilateral triangles. 



Let A, B, C, Sec. fig. 5, be a number of equal and equally 

 attractive particles of a solid so situated that the straight lines 

 A B, B E, E F, F A, form a square. Since the forces of these 

 particles are equal at equal distances, and A E = B F, these 

 particles mutually balance each other ; but if the most minute 

 disturbing force cause the least change in the position of one 

 particle, increase or diminish the force which two mutually exert 

 upon each other, the equilibrium of the whole system is 

 destroyed. Suppose A to be impelled towards E, the force act- 



