THE EQUILIBRIUM OF BODIES IN CONTACT. 481 



SGC" ^ ^ 1 



hence, observing that tan ^ sin -^ = ~ — , we obtain the equation 



lp + n'{^a.+ i)| sec' ^ - r '^^^ '\ a{^a!'+a + l)coS gI sec^ = -a|ia + l} ...(50). 



11. The Equilibrium of the circular Arch, the Material 



BEING supposed UNYIELDING AND THE CONTIGUOUS SURFACES 

 mathematically ADJUSTED. 



Let now the force P be supplied by the opposite pressure of an equal 

 semi-arch, then on the hypothesis made, P is a minimum function of ^. 



Therefore, by (48), 



3a (a + 2) , ^ 



sec'i' = ^^-^ L- (51). 



— + a' (2a + 3) 



. _ tan^P {y+"-(2a + 3)}tan i^ 



.-. 4 sin 2^ = — — = -i -— ; —■ ; 



^ sec- ^ 3a (a + 2) 



therefore, by equation (48), 



-. ,r--„ - 2e[ + 2* (52). 



a(a + 2)r^ J ^ ' 



From which equation ^ may be determined. Also by equation (50), 



P = ^J3a(a + 2)COS-'^-aH2a-l-3)S (53). 



whence P is known. 



Also by equations (50) and (51), 



I^X + P^^ _^ ^^^ W (ia_+l)^vi^ 



•••{^^^ + «(i«^ + «+l)cose| = x/{^ + „'(|„ + !)}„(„ + 2); 



