MECHANICS. 



make the force of the body down the 

 plane equal to the friction. Hence, as 

 we have already explained, M W is to 

 WA as the pressure to the friction. 

 But M W represents the pressure on 

 the plane, and, therefore, W A repre- 

 sents the corresponding friction. Let 

 W D be the element of the drawing force 

 which is perpendicular to the plane, and 

 which therefore diminishes the pressure. 

 Since W M is the undiminished pres- 

 sure, and W D the quantity by which 

 the drawing force diminishes it, the 

 effective pressure will be M D. Through 

 D draw D L parallel to W A. The tri- 

 angles M D L and M W A are similar, 

 and therefore, 



MW : WA :: MD : DL. 



And since W A is the friction corre- 

 sponding to the pressure M W, D L 

 must be the friction corresponding to 

 the effective pressure M D. This then 

 is one part of the force which is to be 

 overcome by the element of the drawing 

 force which is parallel to the plane. The 

 other part is the force of the body down 

 the plane, which is represented by K I. 

 From A take A O equal to K I, and 

 draw O B parallel to A M, and to meet 

 D L produced at B. Then in the paral- 

 lelogram A L B O, the side A O is equal 

 to the opposite side L B. Hence, L B 

 represents the force of the body down 

 the plane. This added to the friction 

 p L, gives the whole force D B, which 

 is to be balanced by the element of the 

 drawing force in the direction of the 

 plane. Hence, if W B be drawn, and 

 also B C parallel to D W, it is plain 

 that W B must represent the drawing 

 force, since W C (which is equal to D B), 

 and W D are its elements in the di- 

 rection of the plane and perpendicular 

 to it. The number of inches in the 

 several lines we have here drawn, is 

 equal to the number of pounds in the 

 forces or pressures which they severally 

 represent. 



(29.) Such is the analysis of the pro- 

 blem when the -plane is inclined, and 



force, corresponding to any angle of 

 draught, may be found by drawing a line 

 from W in the direction of the draught, 

 and terminated in the line O B, or its 

 production. The length of this line will 

 represent the quantity of the drawing 

 force. And on the other hand, if the 

 angle of draught corresponding to any 

 given drawing force be required, it is 

 only necessary to inflect from W a line 

 equal to the given drawing force on the 

 line O B, and the direction of this line 

 will be that of the corresponding 

 draught. 



To find the best angle of draught, it 

 is only necessary to find when the draw- 

 ing force is the least possible, which is 

 evidently done by drawing a perpendi- 

 cular, W B, from W, on O B. This 

 will be the least line which can be 

 drawn from W to OB. Also, since the 

 triangles W C B and B C O are similar, 

 the angle B W O is equal to the angle 

 C B O ; and since B C is parallel to 

 M W, and B O to M A, the angle C B O 

 is equal to W M A ; therefore the angle 

 B W O is equal to W M A ; but this 

 last angle is equal to the elevation at 

 which the body would just move down 

 the plane. 



(30.) [The same may be analytically 

 investigated as follows : 



Let e be the elevation of the plane. 

 The force down the plane is W sin. e, 

 and the pressure is W cos. e ; the fric- 

 tion due to this pressure is 



W cos. e tan. X. 



The element of the drawing force per- 

 pendicular to the plane is 



P sin. x, 



and the diminution of the friction due 

 to this is 



P sin. x tan. X. 

 Hence the effective friction is 



(W cos. e P sin. x) tan. X ; 

 and the entire force to be balanced by the 

 element P cos. x of the drawing force, in 

 the direction of the plane, is the sum of 

 this friction, and the force W sin. e down 

 the plane. Hence we have the equation, 



is 

 from which it appears, that the drawing 



W sin. e + (W cos. e - P sin. x) tan. X = P cos. x. 

 Considering P and x variable, let this equation be differentiated, and we obtain 

 - P cos. x tan. Xdx - sin. x tan.XdP = cos.xdP - Psiu.xdP. 

 This is the same differential equation as 

 was obtained in p. 21, and therefore 

 gives the same result x X.] 



(31.) We shall now investigate the 

 limits of the value of the power which 



subject to the effects of friction. Let us 

 first suppose that the power acts in the 

 direction of the plane. 



If the elevation of the plane be not 

 greater than that at which the body will 



is capable of sustaining in equilibrium just move down the plane, and which 

 a given weight upon an inclined plane, we shall in general call X, it is evident 



