4* 



MECHANICS. 



f The resistance, or pressure A C, will, 

 in this case, be the resultant of the 

 weight and power, which will, therefore, 

 be represented by A D and A B, or by 

 B C and A B. (Treatise I. Chap. II.) 

 But the triangle A B C is similar to 

 the triangle L M H, being equiangular 

 with it ; and, therefore, the power will 

 be represented by the height M H, the 

 weight by the base L H, and the pres- 

 sure by the length L M. 

 To express this mathematically, we 



M H 



R L M 



(94.) It is easy to see that the power 

 acts to greatest advantage, when its di- 

 rection is parallel to the plane. This 

 may be established at once by mathe- 

 matical reasoning ; but it is sufficiently 

 plain, from considering, that if the 



parallel to the planes on which they 

 respectively rest. Considering only 

 the inclined plane AB, the power, that 

 is the tension of the string, the weight 

 P and the pressure are represented by 

 BD, BA, and AD, respectively (92). 

 In like manner, considering only the 

 plane B C, the power or tension of the 

 string, the weight W and the pressure 

 are represented by B D, B C, and C D, 

 respectively. The power, or tension of 

 the string, being the same for both 

 planes, it follows that the weights P and 

 W are as the lengths B A and B C, and 

 that the pressures are as the lines AD 

 and C D. 



This may be submitted to the test of 

 experiment, by ascertaining the power 

 P, which supports any given weight W, 

 and measuring the lengths of the planes. 

 They will always be found to be in the 

 same proportion. To determine the 

 proportion of the pressures, let threads 

 be attached to P and W, and brought in 



LllCtlll) AA V/JLAA O 7 



power be directed above the plane, as directions perpendicular to the planes 

 in fig. 73, it is partly spent in \ lifting over fi xe d pulleys, and let weights R' 

 the weight from the plane, or rather m 

 diminishing the pressure, and only 

 partly in drawing 

 on the other han 



it up the plane. If, 

 it be directed below 



the plane, as mfig. 76, it is spent partly 

 in pressing the weight against the plane, 

 and only partly in drawing the weight 

 up the plane. This will be very evi- 

 dent to the student, who has attended 

 to what has been said of the compo- 

 sition of force in Chap. II. of our first 

 Treatise. But if, on the other hand, 

 the power acts parallel to the plane, its 

 whole effect will be spent in drawing 

 the weight up the plane. 



(95.) If a weight on one inclined plane 

 be supported by a power on another, 

 their proportion will be that of the 



Fig. 



lengths of the planes on which they rest. 

 In this case we may consider P and W 

 (fig. 77.) as two weights, sustained on 

 two inclined planes, A B and C B, 

 by the tension of the string which 

 unites them, and which is the com- 

 mon power which sustains both, and 

 which supports each in a direction 



and R be suspended from them, which 

 shall have the same proportion to P 

 and W, as the lines A D and C D have 

 to B A and B C, and upon removing 

 the planes from beneath the weights, 

 they will retain their positions undis- 

 turbed. 



(96.) The principle of virtual veloci- 

 ties may be easily applied to the inclined 

 plane. In Jig. 75, suppose that at the 

 commencement of the motion the weight 

 is at the foot C of the plane, and the 

 power is at the top D of the altitude. 

 Let the power then descend until the 

 weight shall arrive at the top of the 

 plane. It will have descended through 

 a space equal to the length of rope which 

 has passed over \ the pulley, that is, 

 equal to the length of the plane, and at 

 the same time the weight will have 

 ascended through a space equal to the 

 height of the plane ; so that the perpen- 

 dicular spaces, through which the weight 

 and power move in the same time, are 

 the height and length of the plane, and 

 these are, therefore, the proportion of 

 their velocities. But the proportion of 

 the weight to the power is that of the 

 length to the height. Hence, the power 

 and weight are reciprocally as their 

 vertical velocities, which is conformable 

 to the principle of virtual velocities. 



It would not be difficult to show the 

 application of this principle to the other 

 modifications of the inclined plane ; but 

 this instance will serve our present pur- 

 pose. 



