316 Mr Sang on the Construction of Oblique Arches. 



analogy between this curve and the common catenary has already been 

 pointed out. 



In order to trace the side elevation, we must resume equation (0) 

 which, when adapted to the circular arch, is 



~ = tan s. i y/ \-^^) whence 



r. cots , r+tjj.2 — , 



X = nep. log 



■ Vr2 Z2 



= nep log 10. r. cot s log tan j 45° + — j 

 But the equation 



,. r + ^^rr72 . — ^^ 



j' = — nep. log Y^=. — V r^ — :'- 



~ r — V,.2 — ,. 



is just the equation of the tractory, whence 



x" == — nep. log ■ — 



z »• — Vr2_ •' 



is the equation of a curve having its ordinates greater than those of the 



tractory by the quantity V^-^ r-' this curve I have named the compu' 



nion to the tractory, or, on account of the connection which is explained 

 in the paper, and which at once flows from the above, the inverted cate- 

 nary. 



The equation for the end elevation of a joint adapted to the circular 

 arch is 



-, = nep.logJ{;;.^l^^^} -V,W 



which i.s the well known equation of the tractory. This is the character- 

 istic curve of the circular oblique arch : as all tractories are similar to 

 each other, it is easy to make a table of its co-ordinates. 



The preceding equations enable us to obtain any one of the projections 

 of the joint, and are essential to a knowledge of the nature of the diffe- 

 rent curves. They are, however, inconvenient when we wish to ascer- 

 tain the dimensions of the individual arch-stones, and need, for that pur- 

 pose, to know the intersection of the joint witli any one of the lines of 

 pressure. The equation of the development furnishes us with the means 

 of obtaining these points, as well as all the projections, by processes re- 

 markable for their simplicitj'. To find this equation I resume (L) which, 

 adapted to the circular arch, becomes 



