MECHANICS. 



23 



that the power necessary to sustain the 

 weight in equilibrium *has no minor 

 limit, because, without the assistance of 

 any power, the friction alone is sufficient 

 to prevent the descent of the weight. If 

 the weight be represented by the length 

 of the plane, the pressure is* represented 

 by its base B. Let the proportion of 

 the friction to the pressure, as usual, be 

 /: 1. Hence B/ is the friction. The 

 'force of the weight down the plane is 

 represented by Ihe height H. Hence 

 the force to be overcome by the power 

 in order just to produce motion, is 

 H + B/, which is therefore the major 

 limit of the power which is capable of 

 sustaining equilibrium. Any power 

 greater than this will draw the weight 

 up the plane. 



This may be easily represented by geo- 

 metrical construction. LetAB (fig.lti.) 

 be the inclined plane which represents 

 the weight, AC represents the pressure, 



Fig. 10. 



and B C the force down the plane. Take 

 AD, equal to B C, and from D draw 

 DE perpendicular to the plane, and 

 equal to A C, and from E draw E F, 

 making the angle D E F, equal to the 

 angle X. Hence 



DF :DE::/: i; 



that is, as the friction to the pressure ; 

 and since D E represents the pressure, 

 D F represents the friction. Since A D 

 represents the force of the body down 

 the plane, and D F represents the fric- 

 tion, AF represents a force equal to the 

 combined effects of these, and which 

 would keep them in equilibrium. Any 

 force greater than AF, therefore, will 

 draw the weight up the plane. Hence 

 AF represents the greatest power which 

 can act upon the weight, consistently 

 with equilibrium. 



If we suppose the elevation B A C of 

 the plane to be greater than the angle 



X, the power necessary to sustain the 

 weight will have a minor limit ; for in 

 this case the friction alone is insufficient 

 to prevent the descent of the weight. 

 Upon the principles already explained, 

 the height H expresses the force down 

 the plane, and By is the friction which 

 will resist the descent of the weight ; 

 hence the actual tendency to descend is 

 H B/, which is therefore the minor 

 limit of the power. If F and P" be 

 used in the sense explained in (24), we 

 therefore have 



F = H + B/ 

 P"- H - B/; 



or, following the construction in fig. 10, 

 draw EF', making the angle D E F' 

 equal to X, and D F' will be equal to 

 the friction, and we shall have 



P' = AF, P" = AF. 

 (32.) Let us next suppose that the 

 direction of the power is inclined to the 

 plane. 



The power which acting at any given 

 angle with the plane would just over- 

 come the weight and the friction, was 

 determined by the analysis and con- 

 struction instituted in (28). Hence, 

 if F I, Jig. 9, represent the weight, 

 and WB the direction of the power, 

 the length of the line W B, will ex- 

 press the magnitude of the power 

 which will just overcome the weight 

 and friction ; so that any power greater 

 than WB acting in that direction would 

 move the weight up the plane. Hence 

 F=WB. 



To assign the minor limit of the 

 power will be easy, by a slight modifi- 

 cation of the construction. 



In the process described in (28), 

 instead of making the angle WMA 

 equal to X towards the top I of the 

 plane, let it be made, as in fig. 11, to- 

 wards the foot F. Then, as before, W D 

 (fig. 11.) representing the element of 

 the drawing force perpendicular to the 

 plane, DL will represent the friction. 

 Take AO equal to IK, and through O 

 draw a parallel to AIM to meet LD 

 produced at B. The force with which 

 the weight has a tendency to descend 

 on the plane will be the difference be- 

 tween the part LB of the weight re- 

 solved in the direction of the plane and 

 the friction LD, which is DB. This 

 line DB must then be equal to the ele- 

 ment of the power in the direction of 

 the plane ; and since WD is the element 

 perpendicular to the plane, the power 

 must be WB. 



(33.) Divesting the construction of 



