304 Mr MOSELEY, ON THE 



AM, + BM, + Ms 



cos /= — 



{{A' + B' + 1) {Mr' + Mi + Mi)}k ' 

 in which expression M^, M^, M^, and B, are known functions of A. 



Now / must not exceed the limiting angle of resistance. Therefore 

 cos / must not be less than the cosine of that angle. 



On the whole then we have these two conditions necessary to the 

 equilibrium of a mass intersected by a series of planes, under the cir- 

 cumstances supposed. 



1. That the equation 



-VF,%, F^z, » = 0, 



shall involve no possible roots, except such as correspond to the ex- 

 tremities of the line of pressure, or to points where it touches the 

 surface of the mass. 



2. That the fraction 



AM, + BM, + Ms 



shall for all values of A, corresponding to real sections of the mass, 

 be not less than the cosine of that arc, whose tangent is the coefficient 

 of friction. 



The first of these conditions being satisfied, the parts of the mass 

 cannot turn upon one another. The second being satisfied, they can- 

 not slip upon one another. 



We have supposed the whole of the forces impressed upon the 

 system to be known excepting the force P', which has been deter- 

 mined in terms of the rest. The force P' may be supplied by the 

 resistance of a point in a fixed surface, in which case the amount and 

 direction of that resistance will be known. 



