76 ON THE FACE-JOINTS OF OBLIQUE ARCHES. 



B. The tangent plane to the helical surface at B contains 

 the generator AB and the tangent to the helix which passes 

 through Bj that helix being the line of intersection of the 

 helical surface with the cylinder which is the sofl&t of the 

 arch. These two lines determine the tangent plane. As 

 the tangent to the helix at B lies in the tangent plane to 

 the cylinder and makes an angle (f> (which is the angle of 

 skew-back of the soffit) with B6, the point in which it in- 

 tersects the vertical plane of projection, may be found as 

 follows : — 



Draw BE, making an angle (f) with Bb, and make 6E 

 equal iE,; 6E is the vertical trace of the tangent to the helix 

 at B ; 6Ey is, in fact, the position of the tangent line in 

 question when the tangent plane to the cylinder is turned 

 about B b till it becomes horizontal. But the tangent plane 

 to the helical surface at B also contains the line AB, which 

 is parallel to the vertical plane of projection ; consequently 

 EO, parallel to bC, is the vertical trace of the tangent 

 plane. Hence it follows that the line of intersection of this 

 plane with the plane face B b passes through O (in elevation) ; 

 and since it also passes through B, 06 is the vertical pro- 

 jection of the tangent to the face-joint at B. 



Second. — Let any other point G be taken, of which the 

 projections are g and g'. As before, the tangent plane to 

 the helical surface is determined by the generator G F and 

 the tangent line to the helix traced on the soffit through G. 



The tangent to the helix lies in the plane G^ H; so that 

 by drawing ^/ip making the angle (f> with^L and setting 

 off ^'H equal to LA,, the vertical trace of the tangent to 

 the helix in G is found. Hence the vertical trace of the 

 tangent plane to the helical surface at G passes through H 

 and is parallel to g'C, the vertical projection of the gene- 

 rator at G. 



