Mr Sang on the Construction of Oblique Arches. 313 



a^ =: u cos s, y =t V sin * — u; z z=:z\ '' " trrd 



t) = A' sec *, it = ii?tan« — y, « = «/ 



If the equation of the generating curve of the vault, of which EF is the 

 projection, be taken > ay/o'io 



u — ip«=:0 = B 



the same equation will serve as that of the vault itself ; or in rectangular 

 co-ordinates 



X tan s — y — ipsr=0 = B, whence 



C-!) =-. m =-'^ c^^) -"• 



The equation of the plane containing one of the lines of pressure, is 

 X — X=0 = c; whence 



\dxj \dyf \dz/ r,hp. oi hassoi 



so that the equations of the straight line touching B := , c = are 



—([,'z — 1 ^ ^ 



where X, Y, Z belong to any point in the tangent ; x, y, z to the point of 

 contact. 



' Again, let u — ^ v = = E be the equation of the horizontal projec-^ 

 tion of a joint, or in rectangular co-ordinates, 



X tan s — y — i{x sec s) = 0= E ; then 



I T— 1 = tan s — sec s. r v ; I - — I =: — 1 ; I ^— I = 0- 



\d xj yi y/ V<^ ^/ 



The equations of the joint are B =: 0, c = 0, therefore, those of a line 

 tangent to it are 



X — X- _ Y — y _ Z — .- 



ip' ; "" cp' z (tan s — sec s. f" r) ~ sec s. 6' v • • '^ ) 



The stability of the structure demands, that the line whose equations 

 are (F) be perpendicular to that whose equations are (D), therefore the 

 condition of stability is contained in this equation, 



(f ' z)" (sin « — f v) =. I' V 

 or ^'^^ = \/{sin'-!^'J 



C r = sin « 



(G) 



1 + (?> zY 

 The last form may also be put thus : - ,7, no^:a'>'f 



