VOL. LXXVIII.] PHILOSOPHICAL TRANSACTIONS. 385 



1.3. The increment (pp) of the space divided by the velocity v, is ultimately 

 as the increment of the time, and = the increment of the velocity (v) divided 



by the force — X — in the direction of the tanerent, that is, — = ^?^/^ ^ '^ ; 



■^ PV SP & » ' V 2v* X PY ' 



for pjb substitute , and there results ^^^ = ^ ^ , ; and conse- 



r py ' PY X V 2v^ X py 



quently — = -; but —- = sy := p, hence ^^ = -, andv= -, where 



^ •' SP X PV V 2po ' p v' p' 



a is an invariable quantity. 



Corol. Since v X p, that is, sy the perpendicular multiplied into the velocity 

 (which is ultimately as p/j the space described in a given time) is ultimately as 

 the areas described round the centre s in a given time; but this rectangle = a, a 

 given quantity; therefore the area, described round the centre of force s in a 

 given time, will be a given quantity, and thence in unequal times will be pro- 

 portional to the times. 



1.4. The sagitta qr is ultimately as the force, when the time is given; and 

 when the time is not given, it will be as the force into the square of the time; 

 from which expression, by substituting for aR and the time their values, may be 

 deduced several others. Sir Isaac Newton has demonstrated this proposition with 

 the greatest simplicity; and this is given to show that the same proposition may 

 be deduced from different principles. 



Prop. 1. — 1. Fig. 3. Given the relation between sp' the distance from a point 

 s, and sy' a perpendicular from the point s to p'y, a line touching a curve in the 

 point p'; to find the relation between sp' and sy' (a perpendicular from the point 

 s to p'yt a line touching the curve in the point p) ; in which two curves pp'l 

 and pp'l, the forces and velocities at any equal distances sp and sp are equal, and 

 consequently the perpendiculars sy and sy, at the above-mentioned equal dis- 

 tances SP and sp, are to each other in a given ratio n : w. In the equation ex- 

 pressing the relation between sp' and sy', for sp' and sy' write respectively sp and 

 ^^ ^ ^ , and there results the equation sought : for the distances sp and sp' being 

 equal, the perpendiculars sy' and s^ are as n : w. 



Exam. 1. Let s be the focus of a conic section, then will ^c^ X — .— = sy* 



T ± D 



=3 p% where t and c denote its transverse and conjugate axes, and d the distance 

 sp ; for p write - X p, and there results the equation 4-0^ X — q:— = -7 X p% 

 which is an equation to a conic section of the same name (viz. ellipse, parabola, 

 or hyperbola) as the given curve, of which the transverse axis is t, and conju- 

 gate = , and perpendicular from the focus to the tangent = p. If t and 



c are infinite, and consequently the curve a parabola, and the equation ^Ij X n 

 = p% then will the latus rectum of the resulting equation be Lii^ , 



vol. XVI. 3 D 



