156 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[May, 



ig.l. 



Fig. 2. 



ii m 



Let L S I and M s ni be two square sections at right angles to the 

 axis, and passing through the points S and s respectively. 



b K=6 which is called the obliquity of the bridge. 



a= axil length (see Buck on oblique arches). 



*■ R=^E F, E G radii of the semicylinder respectively. 



w=H N, H=3-14, &c. 



V « V m=X and V S V Z=X'. 



Let X, y, z, be the rectangular co-ordinates of the point s, origin at 

 H, Z measured vertically, and x on the line H K. 



First for this point s we have the following equations : 



a: = rco3. X, y = n-\-°'-^, Z=r Sin. X (V) 

 11 ^ ^ 



These equations are too obvious to need any further explanation. 



Next, to find the equation of the elliptical face a F 6, let x' y' z' be 



the rectangular co-ordinates of any point iu this face, referred to the 



same axis and origin as before, Y the angle made by a section passing 



through this point parallel to the section L S I 



(1 + COS. Y) 6 



. y' = rcos. X,* y' = ■ 



, z' = r sin, X 



(2) 



* This equation is obtained m the following manner :— 

 S'' = tan. z E H X (,• + r los. Y) but tan. .d E o H = .,t 



y = 



_(l+cos. V)& 



as stated above. 



At the point of intersection of these two lines, namely, the spiral 

 and the ellipse, which is formed in the skew elevation, we must have 

 the following obvious conditions: — 



x = x', y=y', z=^z' .•.X = Y 



,, ., a X (1 + cos. Y) h 

 Consequently w+ — = 



By reduction we have x—- — cos. X = (6— 2w) — (3) 



2 o 2 a 



the equation for the intrado. 



The equation to the exfrado is a similar one to that of the intrado, 



viz. — 



X'-^ COS. X' = (6'-2?b') — (4) 

 2 a ^ 2a 



where 6' = 6 + 2 K K' and 7v' = w + ¥.'K 



.•.6'-2j»' = 6-2j!; + 2K K'-2 KK' = 6-2 w 



by similar triangles 2r : (R-r) : : 6 : K K' 



R6-br ,. 6R 



.2KK' 



6'=" 



r r 



Substituting these values in equation (4) we shall have for the 

 equation of the extrado — 



X 



, - COS. X' = (5-2 7») — (5) 

 r ^ 2 o 



Now it is easily seen from equations (3) and (5> that the following 

 relation subsists : — 



(X'-X)- 



n b 



^R COS. X' — r cos. X 



(6) 



If we assume X any arbitrary quantity along the intrado of the 

 skew elevation, the solution of equation (6) will give us X' the 

 corresponding extrado. 



But it is easily seen that this equation cannot, by the present state 

 of analytical science, be generally solved. The best mode of eftecting 

 an approximate solution is by the aid of Lagrange's theorem, as given 

 by Lacroix (see his calculus differential and integral); but this will be 

 rendered unnecessary by the aid of the following proposition: — 



To determine the point O, where the line joining the points S s, 

 intersects the vertical radius, draw s /, S T, perpendicular to G E 

 respectively, and call \/a b K^9 



i s = r cos. X X cosec. S and T S= R cos. X' X cosec. 6 



by similar triangles TS:/s::TO:;0 



that is <o = < E + E 0=?-i^^=^X(T Ex E O) 



by substitution and transposition we have as follows : — 



r- ^ T, /Sin. X' .COS. X— Sin. X . cos.X'\ 



E O =R xrl jr — 1 



\ R COS. X'— r COS. X / 



= RrX; 



R COS. X'—T COS. X 

 Sin. (X'-X) 



(7) 



■R COS. X' — r COS. X 

 bv substituting the value of (6) in (7) we have — 

 ■X) 



_ „ nbn Sin. (X' 

 EO=-^ X 



(8) 



2 a '^ X — X 



It will readily be seen from equations (3) and (5) that X' is always 



greater than X between the limits X =0 and X = ; but (X' — X)con- 



stantly diminishes from X=0, then a maximum to X= where 



(X' — X) =0. From this it appears that E O is not theoretically a 

 constant quantity; but in the construction of bridges (X' — X) when at its 

 maximum is so small,* that we may safely infer Sin. (X' — X) = X — X, 



.-.--'-^ <» 



a constant quantity : this is the greatest value that E O can possibly- 

 attain. 



* All the lines which are made use of in (he foregoing inveslijation will, 

 I think, be perfectly understood by those wlio have read the first chapter in. 

 Mr. Buck's Treatise on tlieSkew Arch. 



