202 Prof. Moseley's Reply to Mr. Earnshaw's 



directions of Q and Q' be those which the surfaces will sup- 

 ply nearest to the direction of PA. 



Now, as is shown in the note, there is a certain direction be- 

 tween which and the normal at either point, if any force be 

 applied, the surfaces will supply a resistance opposite to that 

 force, but if the force be applied further from the normal 

 than this direction, then no resistance will be afforded by the 

 surfaces in an opposite direction. The angle which this di- 

 rection makes with the normal may be called the limiting an- 

 gle of resistance. The resistances Q and Q' will manifestly 

 have their directions inclined to PA at the least possible angles, 

 when they are actually in the directions spoken of above, and 

 make each with the normal at its point of application an angle 

 equal to the limiting angle of resistance. Such, then, by the 

 principle of least pressure, are the actual directions of the 

 pressures at Q and Q'. 



Now let us consider what are the conditions of the equili- 

 brium resulting from this conclusion. 



Let a. = the limiting angle of resistance, 

 2 i = the angle A of the wedge. 

 The angle which Q makes with the side of the wedge is 



Hence, therefore, the angle Qm A, which it makes with PA, is 



2 -"-'• 

 Hence, therefore, the resolved part of Q in the direction PA 

 is Q . sin (« + »)> 



and die wedge being symmetrical about PA, the resolved part 

 of Q is the same. Hence 



2Qsin (a + i) = P; 



... Q= P • 



^ 2sin(a + i) 



point of contact. Let the coefficient of friction equal the tangent of a cer- 

 tain angle, which call a.. Therefore the actual amount of the friction, be- 

 ing the product of the perpendicular pressure by the coefficient of friction, 

 is represented by p cos 6 tan a.. Now this power of resistance acts in a 

 direction parallel to the surfaces at their point of contact, and in a direc- 

 tion opposite to the horizontal part of the force p, which is p sin 6. 

 Hence, therefore, there will be an equilibrium or not, according as 

 p sin & is, or is not, less than p cos 6 tan a, 



or according as tan 6 tan «, 



The surfaces will, therefore, not supply a resistance whose direction is 

 inclined to the normal at an angle 6, greater than that angle a, whose tan- 

 gent is the coefficient of friction. 



