llustrated by
assuming that the arching material is composed of cubes of polished
marble set one vertically above the other in close columns. There would
then be absolutely no side thrust, but, likewise, no arching properties
would be developed, and an indefinite height would probably be reached
above the tunnel roof before friction enough would be developed to cause
it to relieve the structure of any part of its load. Conversely, if it
be assumed that the superadjacent material is composed of large bowling
balls, interlocking with some degree of regularity, it can be seen that
those above will form themselves into an arch over the "centering" made
up of those supported directly by the roof of the structure, thus
relieving the structure of any load except that due to this "centering."

If, now, the line, _A B_, in Fig. 4, be drawn so as to form with _A C_
the angle, [beta], to be noted later, and it be assumed that it measures
the area of pressure against _A C_, and if the line, _C F_, be drawn,
forming with _C G_, the angle, [alpha], noted above, then _G F_ can be
reduced in some measure by reason of the increase of _G C_ to _C B_,
because the side thrust above the line, _B C_, has slightly diminished
the loading above. The writer makes the arbitrary assumption that this
decrease in _G F_ should equal 20% of _B C_ = _F D_{1}_. If, then, the
line, _B D_{1}_ be drawn, it is conceded that all the material within
the area, _A B D_{1} G C A_, causes direct pressure against or upon the
structure, _G C A_, the vertical lines being the ordinates of pressure
due to weight, and the horizontal lines (qualified by certain ratios)
being the abscissas of pressure due to thrust. An extreme measurement of
this area of pressure is doubtless approximately more nearly a curve
than the straight lines given, and the curve, _A R T I D_{II}_, is
therefore drawn in to give graphically and approximately the safe area
of which any vertical ordinate, multiplied by the weight, gives the
pressure on the roof at that point, and any horizontal line, or
abscissa, divided by the tangent of the angle of repose and multiplied
by the weight per foot, gives the pressure on the side at that point.

[Illustration: FIG. 4.]

The practical conclusion of this whole assumption is that the material
in the area, _F E C B B_{1}_, forms with the equivalent opposite area an
arch reacting against the face, _C B B_{1}_ and that, as heretofore
noted, the lower half (