I78 HINTS RELATIVE TO 



the fame in each direction, we have Rn for the fuftain- 

 ing force, exclufive of friction, and the n part of Pn 

 for the friction; and confequently the fum or difference 

 of thefe is the expreffion for the whole force ; and the 

 Problem may be thus conltru&ed. Take PD to DQ 

 as the preffure to the friction, and join PQ ; on PD 

 defcribe a circle, in which take Dv equal to DQ ; join 

 Pv, and draw RV perpendicular to it: then RV will 

 reprefent the direction and meafure of the whole force 

 when it is the leaf! poffible. 



For DQ and Dv are equal, and confequently nf is 

 equal to Vn; but DQ is the n part of DP, therefore nf 

 or Vn is the n part of Pn ; and confequently R V is equal 

 to the fum or difference of Rn, and the n part of Pn; 

 but RV is the leaf! poffible by conitruction, and there- 

 fore the other is a minimum alfo. For draw any other 

 line Rk meeting RV in k and PD in m; and draw mq, 

 mt, parallel to DQ and Dv ; then the fum or difference 

 of Rm and mt is equal to the fum or difference of Rm, 

 andmq; but the fum or difference of Rm and mt is 

 greater than RV, and therefore the fum or difference of 

 Rn and the n part of Pn is the leaft poffible. 



PROBLEM III. 



Given the weight of the body, the inclination of the 

 plane, and the force fuftaining the body in a given 

 dirt ft ion : to find the ratio of the friction to the 

 preffure. 



Take PR as before, (fee Fig. 1.2.) draw Rm in the 

 given direction, and take PR to Rm as the weight of 

 the body to the force fuftaining it ; draw Pm meeting 

 AB in Q, and PD perpendicular to AB; then PD is 

 to DQ as the preffure to the friction. 



PROBLEM 



