in the Faces of Oblique Arches. 



25 



of the arc of cylinder (applying even when the entire barrel or 

 cylinder is built in spiral courses as for an oblique arch) ; and as 

 regards the relation between the angle which the face-plane 

 makes with the axis, and the angle which the direction of the 

 spiral courses makes with the axis (no condition whatever being 

 required for either of them). Only, as the joints are slightly 

 curved, it is proper to suppose that the stones are not very deep, 

 and that the geometrical direction used is a tangent to the lower 

 part of the curve of each : the middle of each joint-curve might 

 be used equally well, but the resulting point of convergence would 

 be slightly altered. 



The theorem may thus be investigated by the processes of 

 analytical geometry. 



Let the diagram represent the horizontal plan of the oblique 



arch, the curved line being the projection of the spiral in which 



one of the longitudinal joints meets the cylindrical intrados, or 



a concentric cylinder (as that which passes through the middle 



of the stones' depth). Let be the origin of coordinates, zthc 



ordinate parallel to the axis, of any point in the helical surface 



which forms the longitudinal joint; as the horizontal ordinate 



transversal to the axis, of the same point, x being not necessarily 



terminated in the curved line ; y the vertical ordinate, its foot 



being in the horizontal plane passing through the axis of the 



cylinder. And let r be the distance of the same point from the 



axis of the cylinder ; 6 the inclination of r to the vertical. Also 



let a be the angle at which the spiral intersects the ridge-line of 



the cylinder, /3 the angle at which the face of the arch cuts the 



same line. Then 



x 

 x = r sin 6, y = r cos 6. and - = tan 6. 



Now if H be the value of 6 in the helical surface when z=Q 

 (II having a different numerical value for every different helix, and 

 being the characteristic of the particular longitudinal joint under 

 consideration), 6 will = H ~n.z ; where n is a constant depending 

 on the elope of the spiral, to be expressed more conveniently here- 

 after. It will be remarked here that the attribution of a con- 

 stant value to H implies that, in any section of the helical sur- 



