184 Mr. Emmet t on the [Sept. 



For by lemma 2, if two such solids be brought into contact, 

 an indefinitely minute portion of the surface of each coincides 

 with an equal one of that of the other ; and by prop. 1, the 

 attraction between these will be indefinitely greater when in 

 contact with each other than when at the least finite distance. 



Cor. Hence, in all cases, the force of cohesion vanishes when 

 the attracting particles are removed to the least distance from 

 contact. 



Prop. 4. — According to whatever power of the distance the 

 centripetal force vary, and whatever be the figure of the particles 

 of matter, the force of cohesion can only be produced when they 

 are in the closest possible contact. 



Let A E B, fig. 7, be an attracting sphere, P, a corpuscle 

 placed without it; fromP draw the straight line, P A B, passing 

 through the centre : in the circumference, A E B, take any 

 points E, m, and join P E and P m ; take E e, m n, two evanes- 

 cent portions of surface, and with centre, P, describe the circular 

 arcs, E F, ej) mo, up, and let F/ = op. It is demonstrated, 

 Princip. lib. 1. prop. 79, that the force with which the sohds 

 formed by the revolution of the figures, E /', 7?i p, upon P B, 

 attract the corpuscle, P, will be as D E^ x F f x ratio of the 

 force with which a point in F attracts the corpuscle, and ? /«- x 

 op X ratio of the force with which a point in o attracts P. Let 



the law of centripetal force be , ^, and since V f = o p, 



^ {Distance}" -^ ^ 



the force with which P is attracted by the solid Ey": that with 

 which it is attracted by mp :: -rrrr- : t; — , let E coincide with A, 



D E* 8 m" 



and the distance P A vanish, then the ratio 7^-=?- : 7;— becomes 



' P F" P 0" 



that of —^ : -— ;;, or of infinity : -p^ ; but in every other position 



of E (P being in contact with A), P E will have a finite ratio to 

 every other finite distance P m ; hence the forces of all other 

 parts of the sphere will have a finite ratio to each other ; and if 

 P A have a finiteratio to P B, the forces of all such figures as 

 E F will also have to each other a finite ratio ; and if P A vary, 

 the variations in the force of each similarly formed sohd Ey"will 



D E" 

 have a finite ratio to each other, since representing the 



force, if P E vary, if its variations have to its original value a 

 finite ratio, the force cannot be indefinitely increased or dimi- 

 nished by a finite variation of P A, excepting in the case when 

 P A = 0. Consequently the phenomena of cohesion can only 

 be produced when the paiticles are brought into actual contact 

 with each other, and in every case arises from the mutual attrac- 

 tion of the portions of their surfaces which then coincide with 

 each other only. The same may be proved in a similar manner 

 of all other sohds. 



