294 PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 



or dividing both sides of this inequality by P cos 0, according as 



tan is less or greater thany. 

 Let now the angle A Q B equal that angle whose tangent is f, and let it be repre- 

 sented by (p, so that tan <p =/■ 



Substituting this value of/ in the last inequality, it appears that the pressure P will 

 be sustained by the friction of the surfaces of contact, or not, according as 



tan is less or greater than tan (p, 

 that is, according as 



is less or greater than ^, 



or according as A Q P is less or greater than A Q B. 



If the angle A Q B be conceived to revolve about the axis A Q, so 

 that B Q may generate the surface of a cone B Q C, then does this 

 cone evidently possess the properties assigned to the cone of resistance 

 in the commencement of this section. 



If the direction of the pressure P coincide with the surface of the 

 cone, it will be sustained by the friction of the surfaces of contact, but the body to 

 which it is applied will be upon the point of slipping on the other. The state of the 

 equilibrium is then said to be that bordering upon motion. 



If the pressure P admit of being applied only in a given plane, there are but two 

 such states corresponding to those directions of P which coincide with the two inter- 

 sections of the plane with the surface of the cone ; these are the superior and the 

 inferior states bordering upon motion. 



Thus, then, it follows, conversely, that " when any pressure applied to a body 

 moveable upon another which is fixed, is sustained by the resistance of the surfaces 

 of contact of the bodies, and is in either state of the equilibrium bordering upon mo- 

 tion, then is the direction of that pressure, and therefore of the opposite resistance of 

 the surface inclined to the normal at a given angle, that called the limiting angle 

 of resistance*." 



8. If any number of pressures P^, P2, P3, &c. applied in the same plane to a body 

 moveable about a cylindrical axis, be in the state bordering upon motion, then is the 

 direction of the resistance of the axis inclined to its radius, at the point where it 

 intersects its circumference, at an angle equal to the limiting angle of resistance. 

 For let R represent the resultant of Pj, Pg, &c. ; then, since these 

 forces are supposed to be upon the point of causing the axis of the body 

 to turn upon its bearings, their resultant would, if made to replace 

 them, be also upon the point of causing the axis to turn on its bearings. 

 Hence it follows that the direction of this resultant R cannot be through 

 the centre C of the axis ; for if it were, then the axis would be pressed by it in the 



* The principle here stated was first published in the Cambridge Philosophical Transactions, vol. 5, by the 

 author of this paper. 



