FRICTION IN MECHANICS. If] 



tion in n and z; fo that nz added to or taken from Rn, 

 the fum or difference may be a given quantity, or the 

 leaft poffible. To do this, let DS be taken equal to 

 DR, and draw SR parallel to PD meeting PQ in M ; 

 then becaufe Rn is equal to rn, the fum or difference of 

 the quantities aforefaid is rz; and when rz is required to 

 be a given quantity, the queftion is reduced to that par- 

 ticular cafe of the inclinations of Apollonius, in fo- 

 lids, which hasbeenrefolvedby Newton and Barrow: 

 the limits of the Problem, or the mode of drawing the 

 line Rr, fo that the intercepted part rz may be the leaft 

 poffible, may be inveftigated as follows : 



* Suppofeitdone, and Rrztnepofition required, and 

 let Rnm be indefinitely near to Rz, and Mh perpendi- 

 cular to Rz, then by applying the analyfis of the an- 

 cients to the Newtonian doctrine of prime and ultimate 

 ratios, mn is equal to zr ; and if from the center R, 

 with the diftances Rz and Rn, the arcs zv and nt be fup- 

 pofed to be defcribed, vn is equal to zt, and confe- 

 quently tr equal to mv ; but rt : tn : : rh : Mh, and tn : 

 zv::Rr: Rz, andzv: vm:: Mh: hz, whence by com- 

 pounding the proportions, tr : vm : : Rr : rh : Rz : zh 

 and as the two firft terms are equal, the two laft are 

 equal, and confequently Rr : Rz : : zh : rh, and divid- 

 ing Rr : rz : : zh : rz, therefore Rr is equal to zh, and 

 confequently the point h is in an hyperbola, whofe 

 afymptotes are QM and SM produced : but becaufe 

 the angle MhR is a right angle, the point h is alfo in 

 the circumference of a circle ; therefore a line drawn 

 from R to h, the point where the hyperbola and circle 

 interfect, is the polition required. 



In the other cafe, where the refinance arifina from 

 tenacity or cohefion is fuppofed to be as the relative 

 preffure againlt the plane, and the force to overcome it 



the 



* Fig. 6. 



