480 Proceedings of the Royal Society 
faces had no cohesion and no friction. For this we require that 
the thrusts to which each stone is subjected should be in directions 
normal to its several surfaces, and should balance each other. 
Now each arch-stone is subjected to three pressures, — one on 
each of its sides in directions tangent to the curve of the arch, and 
a third, the pressure of the superincumbent mass upon the inner 
end of the stone. 
To put the structure in accordance with the usual supposition, 
we must cause the inner ends of these stones to be dressed with 
horizontal surfaces, in order that the pressures exerted thereon be 
downwards. This being done, the usual investigations would hold 
good, and the intrados for a rectilinear extrados would he a modi- 
fication of the catenary. But the inner ends of the arch-stones 
are never dressed in this way; they are rough-hewn, and made 
parallel to the curve of the arch, and thus the deductions from the 
usual hypothesis are quite inapplicable. 
If we suppose the inner ends to be made parallel to the arch and 
to be frictionless, the load resting upon them would tend to slide 
down the slope, and this tendency must he counteracted by a hori- 
zontal resistance from the adjoining masonry ; this, combined with 
the gravitation of the load, produces a resultant normal to the arch. 
In this way the compression of the arch stones is transmitted un- 
changed along the whole curve, instead of being, as in the former 
case, augmented in proportion to the secant of the inclination ; and 
at the same time the horizontal thrust, instead of being conveyed 
unchanged to the ultimate abutment course and there resisted, is 
distributed through the whole depth of the mason work. On in- 
vestigating that form of the intrados which, on this supposition, 
must suit a horizontal roadway, we obtain a differential equation 
which can only be integrated in somewhat complex series. This 
curve lies inside of the circle which osculates the arch at the vertex, 
while the catenarian curve, resulting from the former hypothesis, 
lies entirely without that circle. Between these two curves, there- 
fore, we may have a variety of intermediates, each of which may 
be brought strictly into accordance with the laws of equilibrium by 
giving to the inner ends of the arch stones an appropriate degree of 
inclination. 
In this way we are at liberty to assume, within reasonable limits, 
