Central Forces. 335 



of r, p (or r sin. -4.,): if A=c'^ r=p at the origin, c— 0, and the 

 particle evidently describes a circle whose centre is at the centre of 

 force, and its radius = the initial value of r; but if 4- is not a right 



angle at the origin and c'^ cosec.-4'=A5 c=0, .'. cosec.%|^ = — —= 



const, at all points of the curve, v^'hich shows the curve to be the loga- 

 rithmic spiral, its centre being at the centre of force, c'^ cosec.'-vj^^A 



gives tan.2^=-^-— — ;^5 hence and because c—0, (5), (6) become 



— tan.-l^dr — tan.-4.rc/r . 



dv= J dt=^ ^ 3 supposmg r to decrease; hence by 



taking the integrals on the supposition that v, t commence when 



R tan. 4^ 



r=R, I have v = tan.-^xA./.— > t=- ^ , X(R'^ — '^') ? or ^= 



, R 



vR2 — 2c'cot.4'Xf, t> = tan.-lxA.Z. , ^ =, or t = 



^ VR'-2&cot.^Xt 



pr-. — —7 X 1 — e"^"*^"'-^ h.l.e = l ; the values of r and v in terms 

 2c^cot.4 V /' ' 



R^tan.4 

 of t will give r, v at any time which is less than ^—, — = the time 



from the extremity of R to the centre of the spiral, as is evident by 

 making r=0 in the value of t, at which time v becomes infinite. If -^ 

 is not a right angle at the origin, but A-c'^, then by (3) r^ tan.^-^^ 



— = const, which shows the curve to be the hyperbolic spiral. (Vince's 



I- 

 Fluxions, prop. 129, ex. 2.) In this case (5), (6) become dv=-y-X 



— dr—dr^ 



— f-i dt^= — T=3 supposing r to decrease ; by taking the integrals on 

 ^" V c 



c' R-r 

 the supposition that t, v commence when r—K, I have v=—-^X r> . * 



R-r c' t r . 



t= — Tii-j or r=R — ^v c, v=~X -, and the time from the 



^/c R R-tx/c 



R 



extremity of R to the centre of the spiral =—7^; hence the values 



V c 



of r, V are easily found at any time which is less than —7^. By ta- 



c' —dr 



king the value of the integral of dv=—7^ X — t~ between r and r in- 

 to ^ \/c r- 



