Principle of Virtual Velocities. 135 



equal to \ i> ; calling this T', the point M will be elevated by 

 a quantity equal to \ v' or \ v ; this being designated by i/', the 

 point L will be elevated by a quantity j- D" or | r ; and this 

 is the space through which the weight p passes in rising ; calling 

 this w, therefore, we shall have, whatever v may be, u = \ T, 

 which gives in the case of an equilibrium 



q : p : : u : z>, 

 :: 1 : 8; 



or generally 



q : p : : 1 : 2, 



n denoting the number of moveable pulleys in the system, which 

 agrees with what has been before shown. 



233. In the screw, when the nut passes over a space equal 

 to the distance between two adjacent threads, or when it is 

 elevated through a height equal to DE, the point to which the 

 power is applied describes a spiral about the axis AB, and rises 

 through a space equal to Z)E, the projection of which spiral upon 

 a horizontal plane is a circle, of which BG is the radius. More- 

 over, these motions of the nut, of the point of application, and its 

 projection are such that if the nut describes one half, one third, or 

 any other part of the distance between two threads, the point of 

 application will describe a similar part of the length of the spiral, 

 and its projection a similar part of a circumference of which 

 BG is the radius ; accordingly, if we call u the height through 

 which the nut rises, and v the arc of a circle described in the 

 same time by the horizontal projection in question, we shall have 



u : v : : DE : circum. BG. 



Now the, direction of the force 9, not being a tangent to the 

 spiral, it is necessary, in order to apply to this case the general law 

 of equilibrium, to consider the projection of a very small arc of the 

 spiral upon the direction of the force q ; but this direction being 

 supposed to be a tangent to the circumference above mentioned, the 

 projection upon this tangent will be very nearly equal to the projec- 

 tion upon the circumference, and one may be taken for the other, 

 when we consider only an infinitely small motion of the nut ; 

 consequently, the ratio of the force q, to that of p, will be given,. 



