178 CELESTIAL MECHANICS. I. Hi. 14. 



I • Tkii-« DB.sinADB fa , -p.^ 



we Lave sin DAC= —^ — 4— r-. and DC = sin 



DAC.AD : but if DC represent the weight mgy AC or 

 AD will be the pressure in the direction AB, which will 



hemq.rrr-:^ •=. ■- — :_~t^= mg- ;:;^ .1 For the same reason the 

 •^ DC sin DAC j^ 



action of m' on m will be m'o*^ — 4-» and since these two 



-^ 



forces must be equal, in the case of equilibrium, we shall 

 have mfz=.my\ which is the well known law of the action 

 of a lever, and which explains how two forces, acting in a 

 parallel direction, may cause reciprocal effects, and ba- 

 lance each other [that is, by calling into action a third 

 force equal to their sum, and acting in a contrary direc- 

 tion]. 



We may next consider the equilibrium of a system of 

 points, m, m\ nf, . . , actuated by any number of forces, 

 and reacting on each other. Let /be the distance of m 

 from m\f' that of tw from m'\ and/'^ the distance of mf 

 from mf^; letp be the reciprocal action of m on m', p' that 

 of m on 7/1 \ p" that of m on m' ; and lastly, let m tS, m' S\ 

 rd' S" , . . , be the forces acting on m, m', and »/', and 5, s', 

 «'', the distances of any fixed points, in the directions of 

 those forces, from the bodies to which they belong. We 

 may consider the point m either as being perfectly at 

 liberty, but held in equiUbrium by means of its own force 

 mS, and the action of the other bodies m , m". . . , or as 

 subject, besides these forces, to the reaction of a surface 

 or a curve to which it may be confined. Now, if ^5 be the 

 variation of 5, and S / that of/ taken with regard to this 

 variatfon only, supposing rri to be fixed ; and if ^^f be 

 the variation of/', supposing m' to be fixed ; jR and R 

 being the reaction of the two surfaces, forming, by their 



