ON THE VACE-JOINTS OF OBLIQUE ARCHES. 75 



by a straight line which revolves uniformly about the axis, 

 at right angles to it, and which at the same time has a uni- 

 form motion along the axis, the ratio of these two motions 

 being constant for all the joints. A face-joint is the line 

 of intersection of one of these helical surfaces with the plane 

 of the face. It will readily be seen that these face-joints 

 are curves ; but the curvature is very slight for the ordinary 

 dimensions and angles of arches. The only joint which 

 coidd be straight would be a vertical one ; and there is none 

 such in the face of an arch. 



Let BC represent the projection of half the face of the 

 arch on a horizontal plane containing the axis, which passes 

 through C, and bg'D the elevation of the soffit on a plane 

 perpendicular to the axis ; it is required to show that the 

 tangents to the face-joints on any points on BGD (G is the 

 point of 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 



