1820.] Mathematical Principles of Chemical Philosophy. 353 



will decrease more rapidly than the centripetal force of the par- 

 ticle, when the distance from the centre is increased ; for if D H M, 

 fig. 2, represent a sphere, draw the radius S D, and produce it 

 indefinitely ; at D draw the tangent P E : let this sphere be sur- 

 rounded by a calorific atmosphere, of which the density is repre- 

 sented by the ordinates of the curve K I L, at the distances 

 represented by their respective abscissae ; similarly describe the 

 curve X T N such that its ordinates shall represent the centripe- 

 tal force at the same distances. By Princip. lib. 2, prop. 20, the 

 pressure upon the whole spherical surface D H M is equal to 

 that of a cylinder whose base equals that surface, whose altitude 

 equals that of the atmosphere, and density varies as the density 

 of the atmosphere. Let, therefore, D A G E represent this cy Un- 

 der; draw B F indefinitely near and parallel to A G, and .'. 

 parallel to D E. Take A R, B Q, D P, respectively equal to 

 A X X A K, B T X B I, D N X D L, and through the points 

 R, Q, P, describe the curve RQ P, whose ordinates shall always 

 be equal to the product of the corresponding ordinates of the 

 curves X T N, K I L ; the pressure of the cylinder B A G F is as 

 the quantity of matter and force of gravity jointly ; i. e. as the 

 volume, density, and force of attraction. Now the quantity of 

 matter is as the base BF(=DE) xAB xAK; wherefore 

 the pressure of this cylinder is as the base BFxABxAKx 

 AX; or as the base BF x ABx AR. Let the cylinder be 

 divided into innumerable evanescent strata, and the entire pres- 

 sure will be as the surface D E x area D A R P ; also the 

 pressure upon any part of the surface will be as that surface x 

 area D A R P, and the force itself as the area D A R P, and at 

 any other distance P', if P' O be drawn perpendicular to A D, 

 the force will be as the area A R O P' ; and since the area x-Y B T X 

 of the curve XTN increases more rapidly than the ordinate AX, 

 (the force varying as any inverse power of the distance from S) 

 much more then will the area A R Q B of the curb R Q P increase 

 more rapidly than the ordinate A X of the curve XTN, but the 

 areas A R Q B, A R O P' are proportional to the compressing 

 forces at B and P', they must be proportional to the elastic force 

 at the same distances. 



Cor. 2. — The density of a calorific atmosphere may be sup- 

 posed to be less in the lower than in the higher altitudes. Let 

 it vary inversely as the centripetal force ; then R Q P becomes 

 a right line parallel to A D, whence the elastic force will decrease 

 less rapidly than the centripetal force : such cases cannot attain 

 in nature. 



Prop. 7. — Caloric being attracted by the particles of ponder- 

 able matter will have a tendency to separate them. 



Let A, B, C, D, fig. 3, be four equal, similar, and equidistant 

 particles of matter ; join AC, B 1), DC: with centre D and 

 radius D O = C O describe the arc O M. By Princip. lib. :2, 

 prop. 20, the elastic force of the calorific atmosphere surround- 



VoL. XVI. N° V. Z 



