Mr Sang on the Construction of Oblique Arches. 305 



The Side Elevation of a vault with uniform voussoii's would 

 exhibit narrower intervals toward the crown, the breadths 

 being proportional to the sines of the obliquities ; hence the 

 side elevation of a skewed arch must present narrow intervals 

 both at the crown and at the abutment, and wider intervals 

 upon the shoulders. The breadths are proportional to the 

 products of the sines by the cosines of the obliquities ; that is, 

 to the sines of twice the obliquities ; and thus the side eleva- 

 tions of those arch-stones which are inclined at 45° will be the 

 broadest. 



The End Elevation, or the projection of a joint upon the 

 plane of the parapet, possesses the very singular property of 

 being entirely independent of the angle of the skew, and of 

 depending alone on the form of the longitudinal section of the 

 vault. This curious fact can very readily be demonstrated. 

 The projection of a right angle upon a plane parallel to one 

 of its sides is always a right angle, and therefore the projec- 

 tion of the joint upon the plane of the parapet must cross the 

 projection of every line of pressure upon the same plane per- 

 pendicularly. But the projections of all the lines of pressure 

 are equal to, and placed side by side with, each other, and are 

 so whatever may be the angle of the skew, so that the delinea- 

 tion of the end elevation of a joint, which requires only the 

 tracing of a line that may cross all these at right angles, will 

 be performed exactly in the same manner whether the bridge 

 be more or less oblique. When the angle of obliquity dimi- 

 nishes to zero, that is, when the bridge becomes right, the end 

 projections of the joints contract into mere points, which points 

 are the commencements, so to speak, of the permanent curves 

 above mentioned. 



The end elevations of the beds of the voussoirs, or rather of 

 the lines formed by the intersection of these beds with the 

 planes containing the lines of pressure, are also normals to the 

 lines of pressure, and must therefore be tangents to the end 



I, projections of the joints. From this it follows, that a short 

 _ portion of a course, or a single arch-stone, is very nearly con- 

 ^ tained between two planes slightly inclined to each other ; 



