A SYSTEM OF PARTICLES. 255 



SECTION II. 



PRESSURE. 



22. Conceive a vertical tube of given dimensions filled with air, 

 of wliich the distance between two consecutive particles is e, and 

 whose law of action is that of the inverse square of the distance ; 

 the psirticles in the tube being acted on as well by tlie external air 

 as by that in tlie tube. It is clear that the force on any particle 

 will be, that of the air in the tube, diminished by tlie force of an 

 equal volume of air of the density of the atmosphere. 



Let 2rt = the length of a side of the section, which, for convenience, 

 siippo.se square; and consider the action on each unit of the base to 

 be equal to tliat on the central unit, which is, however, by no means 

 accurate; then, if hx, Si/, Sz be the co-ordinates of any other particle 

 measured from this, we have the pressure due to the air in the tube 



~'"^{ix' + S7/ + Szy' 



dx being in the direction of tlie length of the tube : 

 but if Lv = xe 



this is —r 2 



(x' + f + ^y 



X, ij, z being respectively the number of particles by which the par- 

 ticle under consideration is distant from the point in the base. 



The limits of this sum are x = - , y = - , and s = - . 



Now whilst IJ and = have any one particular value respectively, x 

 will go through a/l its values ; if then we expand the above function 

 and take the Jtiiiie integral of each term separately, we shall, by 

 summing the resulting series, have the repulsion corresponding to any 

 value of y and z. 



