Mr Sang o« the Constniction of Oblique Arches. 317 



ov u St f • 



r — ■= cse f. sec — whence ~"*' 



da r 



r = c. cse ?. nep. log tan I - + — ) 

 or, observing- (Imt -=z i, and passing to common logarithmic tables, " * 



r = nep. log 10. >• cse ?. log tun | 45° + — ) 



•whence by inversion 



M. s!n s 



log tan 



(-^^) 



from which the values of i can be very easily found ; especially when 

 they correspond to equi-different values of v. 



The expert computer will now perceive at a glance, that all the ope- 

 rations needed to determine the co-ordinates of the various points may 

 now be arranged in a simple tabular form so as to require scarcely anj' 

 figuring. 



I now proceed to tlie Elliptic Oblique Arch. Put r for the horizontal 

 and f for the vertical radius ; the equation of the curve then becomes 

 It- z' 



. ■ -jaoilj ii ■ ' ? 



which takes the place of (B). 



Tliis equation may also be put under the form 

 « = r sin a, .: = g cos a. 

 where « is the inclination of the trammel bar that would trace out the el- 

 lipse ; from this we find 



S r CSC s f , .,1 



V— = J {n — «2) cos K + f sec « y w 



0^- '■ { J 



CSC S £ / ^ \ I 



: — ^J ('•■ — i*) sin »: 1- J- nep log tan I 45 + — J V 



hence 



Otherwise we obtain 



i 



V ' ' cse .•< ( . „ „. r 1 



At first glance it might be thought that this equation gives a new curve ; 

 it is, however, still a double logarithmic, having its parts determined in 

 the manner already described. 



To find the side elevation we have 



VOL. XXVm. NO, LVI, APRIL 1840. 



whence 



