122 Statics. 



Fig.U4. 208. If the direction of the power be horizontal, the angle 

 A'F q, being the complement of BAL, the proportion becomes 



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

 q : p : : sin AEG : sin BAL, 

 Trig. 32. :: ^ : BL; 



that is, when the direction of the power is parallel to the base of the 

 inclined plane, the power is to the weight as the height of the plane to 

 its base. 



From the proportion 



q : p :: sin ABG : cosd'Fq, 



we infer, as a general conclusion, that so much less power is re- 

 quired according as the inclination of the plane is less, and 

 according also as the inclination of the power to the plane is 



less. 



We have said nothing of the point where the direction of the 

 power is to be applied to the body. This point is determined only 

 by the condition, that the direction of the power meet the ver- 

 tical drawn through the centre of gravity of the body in a point 

 from which a perpendicular let fall upon the plane has the con- 

 ditions mentioned, article 1 96, &c. 



We hence see that a homogeneous sphere cannot be sustained 

 upon an inclined plane, except when the direction of the sustain- 

 ing force passes through the centre of the figure, which is at the 

 same time the centre of gravity. 



209. If several powers, instead of one, are opposed to the 

 action of the weight, what we have said respecting the power q, is 

 to be understood of the resultant of these several powers. If the 

 Fig.115.body p, for example, is supported upon an inclined plane by the 

 cojnbined action of a power 9, and of the resistance of a fixed 

 point jB, to which is attached the cord HD q, passing round the 

 body ; through the point of meeting S, of the two cords BH, q D, 

 suppose a line SF drawn so as to bisect the angle formed by the 

 cords. If this line cut a vertical line passing through the centre 

 of gravity in a point F, from which a perpendicular can be let 

 fall upon the plane that shall pass through the point of contact 



