THE EQUILIBRIUM OF BODIES IN CONTACT. 491 



^\nien 9 = we obtain the segmental arch, and P then equals 



9 



If the weight of the arch itself be imagined to be included in that 

 of its loading, that is, in X, and if x be determined on the same hy- 

 pothesis; if, moreover, R be substituted for r, this expression for P will 

 give in every case a useful approximation to its true value. It is a limit 

 which the pressure on the key can never exceed, and to which it ap- 

 proximates more nearly as the radius of the arch is greater in com- 

 parison to its thickness. It possesses, moreover, this advantage to the 

 practical man, that it admits of an easy geometrical construction. 



13. Let us suppose that the arch were supported at its springing on 

 the edge of its joint at the extrados (see Fig. 19). Instead of assuming 

 p, in equation (46), equal to r, we must now assume it equal to P at 

 the springing, since the line of resistance will manifestly pass through 

 the point of support. By this supposition we obtain, taking p = R cos 9, 



P= 



X{R%m*d,-x\~\{R'-r'\\cosQ-co5Q,\+lR\R'-r'\\Q,~Q\svne, 



i?{cos9 - cos 9,} ■•■(69). 



In the case in which 9 = and X = 0, 



P=^\R- r}{i (R + .) I cot I - i (i? + r + J^)|* (70). 



* The author has verified this formula, and a corresponding formula, for the case in 

 which the arch is supported at its springing on the inferior edges of its extreme voussoirs, 

 by experiments of whicli the results were communicated to the Mechanical Section of the 

 British Association of Science, at their Meeting in 1837. 



Vol. VI. Paiit III. SR 



