20 



MECHANICS, 



parallelogram W D B C. The force W 

 is equivalent to two forces expressed in 

 quantity and direction by the sides 

 W D and W C. The part W D tends 

 to raise the body W from the plane, 

 and, therefore, to diminish the pressure 

 and the friction, while the part W C 

 tends to move the body along the plane. 



By the obliquity of the draught ad- 

 vantage is gained and lost. Advantage 

 is gained, because the friction which is 

 to be overcome is diminished by the 

 effect of that element WD of the 

 draught, which acting upwards^lessens 

 the pressure upon the plane. Advan- 

 tage is lost, because the element W C 

 of the draught, which is effective in ad- 

 vancing the body on the plane, is less 

 than the whole draught W B, which 

 would be effective if it acted parallel to 

 the plane. It is found, however, that 

 provided the angle (B W C) of draught 

 does not exceed a certain limit, an ad- 

 vantage on the whole is gained by the 

 obliquity, that is to say, a less force will 

 put the body in motion, and continue 

 that motion than would do so acting 

 parallel to the plane. 



It becomes, therefore, an important 

 problem to determine what that angle 

 of draught is which affords the greatest 

 possible advantage, or with which the 

 smallest power will move the body along 

 the plane. This is very easily solved 

 analytically ; we shall, however, attempt 

 to explain it by geometrical construc- 

 tion, being a more elementary process, 

 though not the most expeditious. 



Let the drawing force as already ex- 

 plained, be represented by W B, and 

 suppose it just sufficient to put the body 

 in motion. The element W C must 

 then be equal to the friction. Let W A 

 represent the quantity of friction which 

 would be produced by the whole weight 



Fig. 8. 



of the body pressing on the plane, that 

 is W/(7). Since W C represents the 

 quantity of friction which remains after 



the pressure is diminished by the upward 

 element WD, it follows that CA must 

 represent the quantity of friction which 

 would be produced by the pressure 

 DW or BC ; and since f expresses the 

 proportion of the friction to the pres- 

 sure generally, we have 



CA : CB ::/ : i. 



It will be recollected, that we have 

 already shewn (pp. 5, 6) that if the plane 

 W A were elevated until the body W B 

 would just be moved down it, the pro- 



Eortion of the height of the plane to its 

 ase would be that of the friction to the 

 pressure. Hence, in this case, the height 

 of the plane \vould have the same ratio 

 to its base as the line C A has to C B ; 

 and, consequently, the right-angled tri- 

 angle included by the height and base 

 of the plane is similar to the triangle 

 A C B, and, therefore, the angle ABC 

 is equal to the elevation of the plane, 

 which would just give motion to the 

 body. The angle B A C is the comple- 

 ment of this angle. Now since W A 

 represents the whole friction of the body 

 undiminished by the obliquity of the 

 draught, and the angle B A W depends 

 on the proportion of the friction to the 

 pressure, these quantities are both inde- 

 pendent of the direction, or length of 

 the line W B, which represents the 

 drawing force, and will, therefore, re- 

 main unaltered, however that drawing 

 force be changed in its direction or 

 length. 



Thus we have obtained a very elegant 

 geometrical construction, by which the 

 force which is just sufficient to move 

 the body at each angle of draught may 

 be determined. From any point W on 

 the plane draw a perpendicular W M, 

 and take any parts, W A and W M, on 

 the plane and the perpendicular which 

 have the ratio of the friction to the 

 pressure, that is, so that 



WA: WM ::/:i. 

 Then if WM be supposed to represent 

 the weight, WA will represent the fric- 

 tion due to the pressure of the entire 

 weight, and the angle WMA will be 

 equal to that elevation, X (p. 6), of the 

 plane at which the force of the weight 

 down the plane would be equal to the 

 friction. We will suppose WM to be 

 taken of such a length that the number 

 of inches in it is equal to the number 

 of pounds in the weight. Then the 

 number of inches in WA will be the 

 number of pounds which would over- 

 come the friction due to the entire 

 weight, or which acting parallel to the 



