in the Faces of Oblique Arches. 27 



These may be considered as coordinates of the point in the 

 face-joint for radius a + Ba, the origin of such coordinates being 

 the point in the face-joint for radius a. 



Now take a new system of coordinates, y being the same as 

 before, X horizontal in the face-plane, Z horizontal and perpen- 

 dicular to the face-plane. Then, by the ordinary formulas, 

 BZ = Sz . sin /3 — &v . cos /3, 

 bX = $z.cos /3 + o>.sin ft 

 we find on substitution of the values above, 

 SZ = 0; 



£X=(sin0.cos/3 + sin0.tan/3.sin/3) da'^—^Ba'i 



&/=(wa-ftanftcos#) &a', as before. 

 And it will be remembered that, on the face, 

 x a 



X= 



sin/S sm/3 " 



In the diagram, in which the semiellipse represents the section 

 by the face-plane of the semicylinder whose radius is a } it is 

 easily seen (by similar triangles) that the distance, from the place 

 where X meets the vertical axis, to the place where the face-joint 



produced meets the vertical axis, is *<%"> an ^ therefore the 



distance from the centre of the ellipse to the place where the 



face-joint produced meets the vertical axis is 

 cos |3 



X^/ 



-y> or 



, .sin 0.^-^ (rca-ftan ft cos 0) — a cos 0. or wa 2 .cotan B. 

 sin (3 sin v ' ' r 



As this expression is independent of 0, or of the position of the 



face-joint under consideration, it follows that all the face-joints 



produced meet in one point, whose distance below the centre of 



the ellipse is rc« 2 .cotan |8. 



