330 APPENDIX. A. 



and S being =/, we have l.mSds:=:f^u ; consequently 



383. Theorem. If an attractive or re- 

 pulsive force extend to a given distance c 

 among a series of m particles situated at equal 

 distances in a right line, the mutual forces of 

 any two particles being /, and their masses 

 each unity, the tension acting on an obstacle 



at the end of the line u will be -^ — f. 



The number of particles in the line u being w, the num- 



ber acting at any one point will be 2w — ; and when the 



length u is varied, the variation of the distance of the re- 



motest of these particles will be Sm — , while that of the 



w 



particles at a smaller distance will be proportionally smal- 

 ler : and the mean variation of the distances of the par- 

 ticles within the respective spheres of action will be half 

 the extreme variation. For each particle, therefore, th^ 



variation ^mSls will be \ lu — 2c —f =— mfBu, and 

 ^ u u uu 



cc 

 for the whole line, consisting of m particles, w* — fhi, 



' JTlTTtCC 



which, divided by ^u, srives VM^ /. 



•^ ^ uu '^ 



Corollary 1. Hence, if w be given, the tension will 

 vary as the square of the number of particles or density m, 

 and as the square of the extent pf the sphere of action c, 

 conjointly. 



Corollary 2. If there be two forces, a cohesive force 



