u 
obtained an expression from which the eccentricity 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 intersec- 
tion of the soffit with the plane face. The following 
demonstration, besides being shorter than that given by 
the Astronomer Royal; depends only on elementary geome- 
trical principles, and, it is believed, will convey a clearer 
notion of the state of the case, at least to engineers, whose 
point of view for subjects of this kind is frequently different 
from that of the mathematician. 
The coursing joints of an oblique arch are helical surfaces 
of equal pitch ; that is to say, they are surfaces generated 
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 
could be straight would be a vertical one, and there is none 
such in the face of an arch. 
