Note Return to text — DISCUSSION.

In reference to Fig. 16, it was stated that a series of superposed arches or domes were assumed, but that since the reactions of the horizontal laminas of Fig. 20 and corresponding arches were the same, the former were substituted for convenience. If each arch has the vertical thickness, , it will have the same volume and weight, , as the horizontal lamina. The reactions marked on Fig. 20 can be assumed to be those of the corresponding arch, and the resultant of the horizontal and vertical reactions represents the thrust of the arch at the sides; its direction will be that of the tangent line of the arch. When is small, the horizontal reaction is much greater than the vertical one, and the arches are all very flat. Now, considering the whole series of arches, it is plain that the greatest or limiting value of (call it simply ) would be realized if the weight of any arch is entirely held up by the friction on the sides (induced by the lateral thrust) and the cohesion acting there; for then this same condition of affairs would exist for all lower arches,[Footnote 29] and thus is transmitted vertically downward, unchanged, to the tunnel. Stating this condition in algebraic form,