74 MR. J. B. MILLAR ON THE DIRECTIONS OF 



X. On the Directions of the Face-joints of Oblique Arches. 

 By J. B. Millar, B.E. Communicated by Professor 

 O. Reynolds, M.A. 



Eead October 3i9t, 1876. 



It was observed by Mr. Buck, as described in his well- 

 known * Essay on Oblique Bridges,^ that, in the drawings 

 of the faces of oblique arches, when the face-joints were 

 drawn straight lines and produced, they all met in one 

 point on the vertical line intersecting the axis of the arch. 

 That point he called the " focus,'' and its distance from 

 the axis of the arch the " excentricity ." He gives no reason 

 for this convergence of the face- joints; but taking it as a 

 fact, he determined the focus for the joints at the spring- 

 ing, and so obtained an expression from which the excen- 

 tricity may be found for any given arch. 



In the 'Philosophical Magazine' for January 1862, G. 

 B. Airy, Astronomer Royal, demonstrated, by the method 

 of analysis, the truth of the theorem for the tangents to 

 the face-joints at points equally distant from the axis — 

 such, for instance, as the points on the ellipse formed by 

 the intersection of the soffit with the plane face. The fol- 

 lowing demonstration, besides being shorter than that 

 given by the Astronomer Royal, depends only on elemen- 

 tary geometrical principles, and, it is believed, will convey 

 a clearer notion of the state of the case, at least to engi- 

 neers, whose point of view for subjects of this kind is fre- 

 quently different from that of the mathematician. 



The coursing joints of an oblique arch are helical surf aces 

 of equ?,l pitch ; that is to say, they are surfaces generated 



