390 PHILOSOPHICAL TRANSACTIONS. [aNNO l/SS. 



Assume the three equations before given in prop. 4, viz. - = — X / ± -7- 



X/±Sx/"±&c.f, = ^X/±^,X/±^X/'±l.and-"J 



= (^X/±^: X/ ± ^">/"±«'<=- = 5x/±^*f X/ ± «''=•) X a; 

 from the two former may be deduced the equations vv = «/ + (3/ +/« +/'/3 + 

 y, and .. = «!/ + ^3/' +/* +/^' + v, where « = i|^\ p = ± ^^, y= 



+ ic ( 7/ — + ;//'^— + &C.); » = -^; , P = -I- -— -7 — , y = + 4. 



{^—^{- + - — ^-L. + &c.) ; whence may be derived the two equations «/■+ j3/' 

 4-//+/^ + V = «/+/+/-' +/^' + V = tt/ ± p/ ± (T, where ,r = - 



^MP MP'^ '^ — ^M P MP' ^ — MP ' — MP -' 



+ &c. = &c.) X A. Reduce these 2 equations to ], so that/, f", &c. and 

 their fluxions may be exterminated; and there results a fluxional equation of the 

 formula h/ + k/ + l/+ m = 0, where h, k, l, and m, are functions of one of 

 the before-mentioned variable quantities (for example, mp =: w) which may be 

 supposed to flow uniformly, and its fluxion. 



Prop. 7« — 1- Fig- 8« Given the force tending to any point s, the velocity and 

 direction of the body; to find the curve described. 



Let the body acted on by a force/ tending to s, at the distance d' from s, be 

 projected in the direction p'y', with a velocity h ; and let the perpendicular from 

 s to the tangent pV be A; from the general fluent of f X d. where d denotes 

 the distance from s, and /is a function of d, properly corrected, find its velo- 

 city V at distance d, and consequently the perpendicular sy from the centre s to 



the tangent py at distance d = sp, which will be = sy; but a and h are 



given quantities, and v a known function of d; therefore sy and v^(sp^ (d^) — 

 SY^) = PY will be known functions of d; and from the similar triangles spy and 

 PQT may be deduced py : sy :; pt = © : ax, and consequently sp X ax = d X 

 " ^ ^ (which is a known function of d multiplied into d) will be as the incre- 



PY 



ment of the area described round the centre of force, of which the fluent pro- 

 perly corrected is proportional to the area described round the centre of force, 

 and consequently to the time. In like manner, ~~ — - = — (proportional to the 

 increment of the angle described by the body round s) is a function of d mul- 

 tiplied into D, of which the fluent properly corrected, or angle, will be as a func- 

 tion of D. 



1 .2. Fig. 9. Given the above-mentioned force, &c. ; to find an equation ex- 

 pressing the relation between the absciss sm = ar and ordinate mp = y of the 

 curve described, and their fluxions. — From the similar triangles Fpo and lpm 



