THE EQUILIBRIUM OF BODIES IN CONTACT. 483 



•■•'-* =T^i? '="■ 



Suppose the arch to be supported upon upright piers of a given 

 breadth a, (Fig. 18), and let it be required to determine what is the 

 greatest height (s) to which they can be carried. 



By equation (23), 



Px (k — a) cos <1> 

 1 



2 



where k = p — r. 



la^ — P, sin <i>' 



Also by equation (.55), 



r(a + 2)(tan---j 

 p-r = 



{^a+l)0 + {l--}ur)cotO' 



.(59). 



... [ >-(i^- + a). iJ r(a.2)lt^l-lj I 



\l(r-r-{a- ^a')f{{^ a +1)6 + (1-^ a-) cote ]"' "" 



If = T , or the arch be a semi-circle, 



i r° (^ g- + g) TT ' ' ^ ~ 

 irr_ ;.^(„ _ la') 



It is evident, from equation (50), that as X is increased ^ increases ; 

 that is, the points of rupture descend continually upon the arch as it is 

 more loaded. The experiments of Professor Robison on chalk models 

 are explained by this fact*. 



* Having constructed chalk models of the voussoirs of a circular arch, and put them 

 together, he loaded the arch upon its crown, increasing the load until it fell. The first 

 tendency in the chalk to crush was observed at points of the intrados, equidistant from the 

 crown on either side, but near it ; these points were manifestly those where the line of 

 resistance first touched the intrados. As the load was increased, the tendency to crush ex- 

 hibited itself continually at points nmre dislant from the crown — that is, the points of rupture 

 descended. 



Vol.. VI. Paut III. SQ 



