Central Forces. 337 



dr 'I -rdr i r • i 



dt — —7^ X — ; the value oi r m these equations 



is never greater than R, and it is evident that when r=R, it cuts 



the curve at right angles: by taking the integrals of these equations, 



d 

 supposing t, V to commence when r=R, I have v= -Xh.l. 



R-y/c 



R+v'R-^-r-- ^^^^ ^=-— xVRa.-r^ or r = \^R^-cr- (8), t; = 

 r v c 



€' R-f-^v^c^ 



• -Xh.l.-y - (9), r, v are easily found by (8), (9) at any 



R-y/c V R2 — ci^ 



R . . ^ ^ 



time which is less than —7== the time from the extremity of R to 



the centre of the spiral. (7) agrees with Newton's construction of 

 this case of the general question ; Prin. prop. 41, cor. 3, see his fig. 



5. for R= his CV, ^=VCP, r^CT^CP, and /i./.?±^__Zll 



r 



is as the hyperbolic sector VCR. 



If c'^ is >A, c is positive, as is evident by (1) of (3); or if the 



central force should be repulsive c will be positive, for A is negative 



in this case J and the signs of the terms involving A in (1), (3) must 



c'2=fA 

 be changed: hence by putting = R^5 (the sign — being used 



when the force is attractive, and the sign + when it is repulsive.) 



c' dr 1 rdr 



(5), (6) become dv = -t= X — — ? dt=—^ X /- ^ ; the 



vc r-y/r^ — R^ Vc v^r^— R^ 



least value which r can have in these equations is evidently R, and 



it is evident that r cuts the curve at right angles when it =R; hence, 



supposing that v, t commence when 7-=R, by taking the integrals, 1 



have v=— — =-Xarc sec. = - ), ?!=\/!LZ — 1, or r=\^ct^ -{-R^y 

 R-y/c \ K/ c 



v=- — = X arc I sec. = ( it | , whence r, v are easily found 



R^/c V \ R // • ^ 



at any time; also r=RXsec.[ '^^t;\, (10). 



(10) agrees with Newton's construcdon ; Prin. prop. 41, cor. 3, 



see his fig. 5. for R= his CV, v= ang. VCP, r=CT=CP, ^^^ v is 



c' 



Vol. XXL— No. 2. 43 



