480 PROFESSOR MOSELEY, ON THE THEORY OF 



10. The Circular Arch. 

 Let the intrados and extrados be circular cylindrical surfaces ; 



.-. f" [%•'' sin dciedr = - i(^ - r%cos G - cos 9), 



sm6 f''j\d0dr = ^ sine (E'-r')(9-e) ; 



.-. ^(R' - r)(cos e - cos 6) + Xx + Pp 



= p\Xsme + Pcose+\{R'-f'){e-9i)%m9\ (46). 



Therefore, by equation (43), 



^(R'- r%cos e - cos i') + Xx + Pp 



= r{Xsin^ + Pcos^ + i(£'-r^)(4'-e)sin4'} (47). 



(It 

 By equation (44), observing that -r- = 0, 



^(/f - r')sin* = r{Xcost-Psin ^ -t- 1(^' -r-)(* - G)cos* + i(^=-r') sin^} ; 

 hence, assuming R = r(l+a) 



(f* Try \ f/y \r- > 

 — + d'(2a + 3)ltan* = |— p-3a(a + 2)ei + 3a(a + 2)S' (48). 



By equation (47), 

 P {p — rcos"^] 



= {Xr+^r{R'-r^){-i' -e)\sm^-Xx+^R-r'){cos^-cose) (49). 



Also, by equation (48), 



Xr + ^r(R-t^){^-e)={Pr+lrT^a'(2a + 3)}ta.n<i'', 

 .-. P\p-rcos^\ = {Pr+lrV(2a+3)} tan ^ sin ^ 



— Xo; + r'a (^a' + a + l)(cos ^ - cos 9) ; 



