MECHANICS. 



41 



exactly the same proportion as the 

 weight W and the power P. The 

 quantity of the resistance may be de- 

 termined, experimentally, in the same 

 manner. Let a string be attached to 

 the weight, and brought in a direc- 

 tion A B, over a fixed pulley, and 

 let a weight R be suspended from it, 

 which will bear the same ratio to W 

 and P, as the line A B or D C bears 

 to the lines A D and A C. Upon re- 

 moving the plane, it will be found that 

 the weight A will remain suspended, 

 undisturbed in its position. Hence, it 

 appears that the tension of the string 

 A C, which is a force equal to the 

 weight R, supplies the place of the 

 plane, and produces the same mecha- 

 nical effect. The force R, therefore, 

 is the amount of the pressure of the 

 weight upon the plane*. 



Since all equiangular triangles have 

 their sides proportionalf, it follows that 

 if any triangle be drawn, whose angles 

 are equal to those of a triangle, one 

 side of which is vertical, another per- 

 pendicular to the plane, and the third 

 in the direction of the power, the sides 

 of that triangle will always be pro- 

 portional to the power, the weight, and 

 the pressure upon the plane. This re- 

 lation between the power, weight, and 

 pressure may be very simply expressed 

 mathematically. Let the angle formed 

 by the vertical line A D, (Jig. 73.) and 

 the perpendicular A B to the plane be 

 A, and let the angle B D A, formed by 

 the vertical line A D, and the direction 

 A C of the power be D ; and the angle 

 under the direction of the power A C, 

 and the perpendicular to the plane, viz. 

 D B A be B. Since the sides of tri- 

 angles are proportional to the sides of 

 the opposite angles, we have 



?_ Sin. A JR = 'Sin^D 

 W " Sin. B W Sin. B ' 



(92.) In the preceding investigation 

 of the proportion of the power, weight, 

 and pressure, we have conceived the 

 power to act in any direction whatever. 

 If it act in the direction of the plane, 

 the triangle whose sides will determine 

 its proportion to the weight, will be A 

 C B, (fig. 74.) in which B C is the di- 

 rection of the power, A B of the weight, 

 and A C of the pressure. This triangle 



* The experiments which we have thought it ad- 

 visable to describe, in verification of the theory, are 

 not those which are the most easily executed, but 

 those which we conceive to be best adapted to ren- 

 der the theory intelligible, independent of much ma- 

 thematical reasoning. 



f See D ABBEY'S Popular Geometry, Art. 112. 



Fig. 74. 



is evidently similar to the triangle F D 

 E, formed by the height, length, and 

 base of the plane. Hence, in this case, 

 the height D F of the plane represents 

 the power, the length, the weight, and 

 the base of the pressure. 



This may be verified experimentally. 

 Let a thread, attached to the weight A, 

 be brought parallel to the plane, and 

 passed over a fixed pulley at D let 



Fig.15 



such a weight P be suspended from it, 

 as will bear the same proportion to A 

 as the height D E bears to the length 

 D C, and it will be found to sustain the 

 weight. The amount of the pressure 

 may be shewn to be represented by the 

 base C E in a manner exactly similar 

 to that explained in/Sgr. 73. 



This may be expressed mathema- 

 tically thus : let E be the elevation of 

 the plane 

 P DE R CE 



(93.) If the power act in a horizontal 

 direction, or parallel to the base of the 

 plane, its proportion to the weight will 

 be that of the height of the plane to the 

 base. For, in Jig. 76, let A B be ver- 



Fig. 76. 



tical, and, therefore, in the direction of 

 the weight ; and let A D, parallel to 

 the base, be in the direction of the 

 power, and A C, perpendicular to the 

 plane, will be in the direction of the re- 

 sistance. 



