THE EQUILIBRIUM OF BODIES IN CONTACT. 479 



Therefore by the condition of the equality of moments. 

 J^ £ r'sme(lOdr + Xx + Pp = fj{Pcose+Xiim0 + sme f f^rffedr\....(4>2). 



At tlie pohit of rupture the line of resistance meets the intrados ; 

 therefore at this point 



/> = '• (43). 



Also, generally, sufficient dimensions of the arch being supposed, 

 the line of resistance touches the intrados at this point ; 



.(44). 



dp _ dr 



'■ Tie ^dd 



Moreover, R = F0, r = fQ (45), 



these being given functions of 0. 



Assuming 4' to be the value of Q at the point of rupture, and sub- 

 stituting it for f? in the five preceding equations, we may eliminate 

 between them the four quantities p, E, r, j}, or the four p, R, r, *. 

 There will result an equation involving the quantities P and -V in the 

 one case, and P and p in the other. 



Now if the force P be supplied by the pressure of another opposite 

 and equal semi-arch, it has been shewn, (see Memoir on the Theory of 

 the Arch, Vol. V. Part iii.) that if the masonry be supposed perfect, P 

 is a minimum in respect to the variable p ; moreover if the masonry 

 be supposed to possess those yielding properties which obtain in practice, 

 and which shew themselves hi the settlement of the arch, t\\enp=R\ 

 according to either of these conditions, P and ^ may therefore be de- 

 termined. 



The values of P and p becoming thus known, they may be substi- 

 tuted in equation (42), and the equation to the line of resistance will 

 thus be completely determined. 



