294 
PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 
or dividing both sides of this inequality by P cos 0, according as 
tan 6 is less or greater than f. 
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 -f- 
Substituting this value of f 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 6 is less or greater than tan <p, 
that is, according as 
6 is less or greater than <p, 
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 1? P 2 , P 3 , &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 P l5 P 2 , &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. 
