Friction. 1 49 



resultant of the power and weight should be directed according 

 to FI. (2.) That the point /, where the line FI meets the plane, 

 should belong to some point of the base DE ; otherwise the body 

 would tend to turn. 



This being premised, we have 



p : q : : sin q FI : sin p FI, 

 or, letting fall upon the plane the perpendicular FH, 



p : q :: sin (qFH HFI} : sin ( P FH + HFI). 



Now the angle HFI is the complement of the angle of friction; 

 and the angles q FH, p FH, are supposed to be known, since the 

 the direction of the power is considered as known, together with 

 the inclination of the plane, which is equal to the angle p FH ; Geom. 

 we have accordingly the ratio of p to q. 



If we would determine this ratio in lines, we have only to 

 draw through any point B of the inclined plane, the line BT, 

 making with AB the angle ABT = HF q, and the line BV, mak- 

 ing with AB the angle ABV = HFI, the complement of the 

 angle of friction. Then drawing the horizontal line AT, we shall 

 have 



p : q : : VT i BT, 



since the angle VBT - ABT ABV = HF q HFI, and the Ge om. 

 angle BVT = BAY + ABV = p FH + HFI. Now in the 78 - 

 triangle BVT 



VT : BT :: sin VBT : sin BVT. Trig<32v 



Instead of making the angle ABT = HF q, and the angle 

 ABV = HFI, we may draw BT perpendicular to the direction 

 of the power q, and BV perpendicular to FI\ this amounts to 

 the same thing, and is moreover analogous to the course pursued 

 in article 203. 



245. The second condition shown to be necessary in order 

 that the power q may be upon the point of moving the body, 

 renders it evident, that when the body does not rest upon a point, 

 the direction of the power produced must meet the vertical, 

 drawn through the centre of gravity, at the point F, where thisFig.132. 

 last line is met by the line IF, proceeding from some point of con- 



