Let BC represent the projection of half the face of the 
arch on a horizontal plane containing the axis, which passes 
through C, and hg’I) the elevation of the soffit on a plane 
perpendicular to the axis ; it is required to show that the 
tangents to the face joints at any points on BGD (G is the 
point on the soffit of which gg are the projections) meet at 
one point on the vertical line CA. 
The tangent to the face joint at any point is the line of 
intersection of the plane face and the tangent plane to the 
helical surface at that point. 
First. — To find the tangent to the face joint at the point 
B. The tangent plane to the helical surface at B contains 
the generator AB and the tangent to the helix which passes 
through B, that helix being the line of intersection of the 
helical surface with the cylinder which is the soffit 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 (p, which is the angle of skew- 
back of the soffit, with B5, the point in which it intersects 
the vertical plane of projection may be found as follows : — 
Draw BE^ making an angle <p with Bb, and make 6E equal 
; E is the vertical trace of the tangent to the helix at B 
— BE^ is, in fact, the position of the tangent line in question 
when the tangent plane to the cylinder is turned about Bh 
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 60, is the vertical trace of the tangent plane. Hence it 
follows that the line of intersection of this plane with the 
plane face B6 passes through 0 (in elevation), and since it 
also passes through B, 06 is the vertical projection 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 GF and 
