EQUILIBRIUM OF THE ARCH. 805 



If, however, there enter two or more resistances of surfaces among 

 the forces which compose the equilibrium, since the magnitudes of 

 these and also their directions may be any whatever, within the limits 

 imposed by the friction of the surfaces; the problem remains, in so 

 far as the known conditions of equilibrium are concerned, indeterminate, 

 and recourse must be had for its solution to other principles. 



7. Suppose the mass AJS to be acted upon by any number of forces 

 among which is the force P being the resultant of certain resistances, 

 supplied by different points in a surface Sb, common to the inter- 

 sected mass and to an immoveable obstacle SC. 



Now it is clear that under these circumstances we may vary the 

 force P', both as to its amount, direction, and point of application, 

 without disturbing the equilibrium, provided only the form and 

 direction of the line of pressure continue to satisfy the conditions im- 

 posed by the equilibrium of the system. 



These are manifestly, that it no where cut the surface of the mass, 

 except at P" and within the space JSb, and that it no where cut a 

 section of the mass or the common surface of the mass and obstacle, 

 at any angle with the perpendicular greater than the limiting angle 

 of resistance. " 



Thus, varying the force P', we may destroy the equilibrium, either, 

 first, by causing the line of pressure to take a direction without the 

 limits prescribed by the resistance of the section through which it 

 passes ; or, secondly, by causing the point P to fall without the surface 

 Bb, in which case no resistance can be opposed to the resultant force 

 acting in that point ; or, thirdly, the point P lying within the surface 

 Bb, we may destroy the equilibrium by causing the line of pressure 

 to cut the surface of the mass somewhere between that point and P'. 



Let us suppose the limits of the variation of P' within which the 

 first two conditions are satisfied, to be known ; and varying it, within 

 those limits, let us consider what may be its least and greatest values 



