EQUILIBRIUM OF THE ARCH. 308 



In the remainder of this paper the angle 0, or tan-'^ will be called 

 the limiting angle of resistance*. 



From the above then it appears, that unless the tangent to the 

 line of pressure at the point where it cuts any section of the mass, 

 make with the perpendicular to the plane of that section an angle, 

 which is not greater than the limiting angle of resistance, the surfaces 

 there in contact will slip upon one another. 



This condition may be expressed analytically as follows : 



% = Ax + By + C 

 is the equation to the plane of any section of the mass, therefore 



x-x^ = - Ai^-z), y-y,= -B {x-z), 



are the equations to the perpendicular to that section. And the angles 

 which that perpendicular makes with the co-ordinate axes have for 

 their cosines 



-A -B -1 



VA^TW+\' VA' + B'+l' VA' + B' + l' 



Also it appears from the given equations (3) to the resultant 

 force, or tangent to the line of pressure, that this line makes angles 

 with the co-ordinate axes which have for their cosines the quantities 



M, M, Ms 



Hence, therefore if / be the inclination of these lines to one 

 another, 



* It is here supposed that the coefficient of friction f is constant for the saifie surfaces, 

 whatever be the force B! by which they are pressed together. This is usually assumed 

 to be the law of friction. It is only however an approximation to that law. The ex- 

 periments of Mr Rennie shew that f must be considered a function of R' increasing con- 

 tinually, but very slowly, up to the limits of abrasion. 



:fs:. 



