THE EQUILIBRIUM OF BODIES IN CONTACT. 485 



the great practical question is to determine the conditions of the pressure 

 under those possible circumstances, which are most unfavourable to the 

 stability of the arch ; circumstances whicli manifestly occur in the state 

 bordering upon its rupture. This question necessarily then supposes a 

 direction of the pressure, and therefore of the line of resistance, touching 

 the extrados at the crown, and the intrados at the liaunches; and, this 

 being supposed, all those conditions of the equilibrium which depend 

 upon the nearer approach of the voussoirs after the first striking of the 

 center, or which arise from the long continued pressure, or from the in- 

 fluence of changes in the temperature, are eliminated. 



Let us then assume that the line of resistance touches the extrados 

 at the crown, so that p = r{\ + a) cos ; by equation (49), 



F 1(1 + «) cos e - cos ^} = {X + r" {\ a' + a){^ - e)} sin ^' 



-h r-{a + a" + ^ a?) (cOS "P - COS G) ;- . 



By equation (48), 



Ptan^ = X + r'(ia' + a)(i'- e) - r^«=(ia + i)tan^; 



(1 + a) cos 6 — COS 'P 



tan ^ 



\-s. + (i «' + «) (^ - ©)| sin '^ -1- (« -f a"' + i «') (cos >!' - cos G) - ^ 

 = ^ — _ 



-p + (h»' -\- a) (* - G) - aMi a -|- ^) tan ^ 



Dividing this equation by sin ^, subtracting unity from both sides, 

 and reducing 



(1 +a) cos 9 - cos •4" — sin ^ tan ^^ 

 tan 4' 



a{^a + ^) tan * sin ^P -h (a + a' + ^a')(C0S ^ - COS 0) ^ 



— + i^a + a)(-^ - e) - a'(^a + ^) tan^ 

 3QS 



