Inclined Plant. 121 



the two directions according to which the same power will sus- 

 tain a given weight upon the same plane, are equally inclined 

 with respect to a perpendicular to this plane, and consequently 

 with respect to this plane itself; and they both fall on the side 

 of a perpendicular to this plane, opposite to that in which the 

 gravity of the body is directed. 



206. In the same proportion, 



q : p : : sin HFp : sin HF q, 



if, instead of the angle HF 'p, we put the inclination ABG of the 

 plane, which is equal to this angle, and instead of sin HFq, its Geom. 

 equal cos A'F ' q, FA' being drawn parallel to BA, we shall 209 ' 

 have 



q : p : : sin ABG : cos A'Fq, 

 and hence 



x 9in 



Therefore, the inclination of the plane and the weight remaining 

 the same, the power q must be so much the smaller, as the cosine 

 of its inclination to the plane is greater ; accordingly, as the 

 greatest of all the cosines is that of 0, we say that the direction's- 1 **- 

 in which a power acts to the greatest advantage, in sustaining a weight 

 upon an inclined plane, is that which is parallel to this plane. 



207. In this case the proportion 



q : p :: sin ABG : cos A'Fq 

 becomes 



q : p : : sin ABG : 1 or radius. 



Now if, from the point A, we let fall the perpendicular AL upon Fig113 

 the horizontal line BG, we shall have in the right-angled triangle 

 ALB, 



therefore 



q ' p :: AL : AB-, 



that is, when the power acts in a direction parallel to the plane ; it 

 to the weight as the height of the plane is to its length. 

 Mech. 16 



