l82 HINTS RELATIVE TO 



the fum of all the RF or PQ, namely, as the area 

 APQG; hence the truth of the propofition is mani- 

 feft. 



Corollary 1. Becaufe Mn or the fluxion of y is to 

 Mm the fluxion of the curve, as MR or PN to RF or 

 PO, therefore if PN be a fundion of AP, PQ will be 

 a fourth proportional to the fluxion of the ordinate, the 

 fluxion of the curve AM, and this function; wherefore 

 if the carves HM and AM be given, the nature of the 

 curve GO will be known, and its area may be found 

 by the common methods of quadratures. 



Corollary 2. It is evident that when the planes are 

 inclined to the horizon, the frictions of the right and 

 curvilinear planes are (till in the fame ratio as in the 

 preceding cafes, and confequently may be found by the 

 fame mode of proceeding. 



Corollary 3. It is alfo evident, that the above me- 

 thod holds good whether the parts of the body are con- 

 nected together or not, with refpetl to their motion in 

 the direction RM, fo long as each elementary part MR 

 may be confidered as fultained at the point M by a 

 force parallel to MP ; but when the body is rigid or 

 inflexible, the cafe becomes more fimple, for MR is 

 then conftant, and APNH becomes a parallelogram. 



Corollary 4. By fuppofing given properties to exift 

 in any two of the curves AM, HN, or GQ, the nature 

 of the third will be known ; and hence a number of 

 problems relative to friction may be propofed and re- 

 folved by a proper application of the direfct andinverfe 

 methods of fluxions. 



PROPOSITION 



