302 Mr. C. Fox on the Construction of Skew Arches. 
It is evident from fig. 7, that if spiral planes are considered 
as composed of spiral lines placed at various distances from 
the centre of the cylinder, each of these lines will form a 
different angle with the axis; and therefore, as an arch has 
always some thickness, that although we have the inner edge 
of the spiral plane placed at right angles to the thrust, yet 
every other portion is gradually departing from a right angle, 
and is, therefore, exerting its force in an improper direction: 
thus an arch of this description can never exert its thrust in 
the direction of the bridge, but is endeavouring to push the 
abutments obliquely. 
To get the thrust strictly correct, I have supposed the arch 
to be cut into two rings of equal thickness (see fig. 8); and 
having considered the external ring as removed, have pro- 
ceeded to develop the outside surface of the remaining one: 
this I shall hereafter speak of as the intermediate develop- 
ment, as it is the development of a surface midway between 
the extrados and soffit or intrados. 
Upon this intermediate development I place the approxi- 
mate line, and then draw all the courses square to it; by which 
means we obtain a line in the centre of each stone exerting its 
force in the true direction, and thus get rid of the disadvan- 
tage of twisted beds to the stones, as in proportion as the one 
half of this bed exerts its force in an oblique direction on 
the one hand, the other half acts in the opposite direc- 
tion, and is therefore always producing a balance of effect, 
which resolves the various forces into one exerting all its 
power in the true direction, which is the object to be ob- 
tained. 
Having explained the mode of setting out the beds of the 
stones, a little may now be said on the situation of the cross- 
joints: by these will be understood the joints between the va- 
rious stones constituting a complete course. 
Where an arch is built of stone throughout, the situation of 
these joints is of minor importance; but where stone is expen- 
sive, it is common to make the faces of the arch only of stone, 
filling in the intermediate space with brick-work; as in these 
instances the cross joints form the boundary between stone- 
and brick-work, it becomes a point of considerable importance. 
This is the case in the Watford viaduct; each stone here is 
equal in thickness to five courses of bricks, so that there are 
five thicknesses of mortar in the brick-work to one in the 
stone. Mortar always is compressed into a smaller compass 
when the centring is struck, and the full weight of the arch 
comes upon it. In consequence of this tendency, that por- 
tion of arches constructed of brick-work, always subsides much 
