22 



Proceedings of tlie Uoi/al Irish Academ'n. 



Next suppose that the rigid arc ^/S'has a hinge at any point B\ let us 

 consider the equilibrium of the upper rigid part AB. At B introduce the 

 horizontal force H equal to the shaded part of the Q parallelogram, and the 

 vertical force F equal to tlie shaded part of the P parallelogram. The three 

 forces T, H, and V balance the rigid arc BA. But these shaded areas are 

 proportional to f/aiid ae respectively, or to cos d and sin d, therefore the 

 resultant of H and V equals T, and is tangential to the rib at B, 

 so that there is no bending stress at the hinge B. But B was any 

 point, and so tlie whole arc AS may be a ^;fc)M<.wi of hinges, that is, 

 it need not be rigid, but free to change its sliape. These resultant stresses 

 (loads) on the back of the rib are not necessarily due to the requirements of 

 a surrounding mass that its own equilibrium be assured. They can be 

 produced artificially, as we shall see when dealing with the vertical arch. 



\, \sunk 



vertical or 

 bitilt and 

 surrounded 

 with -water, 



mud or sand. 



Earth (loose or punned) 

 Fig. \a. 



The result of our latest investigation points dX tv;o imirs of synnnelrical 

 conjugate loads which balance a " complete arch." Indeed, their number 

 may be indefinite, with only two of them real. The pair already discussed 

 are like, wliolly positive and uniform. 



It is most remarkable that there is another pair for the circular rib. 

 They are unlike, but symmetrical, in that each has a central smaller part of 

 opposite sense from the end parts. The corresponding geometrical diagrams 

 are parabolic in form, a sixth of the depth at the vertex being on the opposite 

 side of the base. 



They are shown on fig. 2a. The linear unit is taken as r, the radius of 

 the rib, to simplify the aritlimetic; it makes tlie parabolic segment isosceles, 

 as the two co-ordinates from its vertex to any point x and o^ are equal.' 



