336 Central Forces. 



finite, (supposing V to commence when r= infinity,) I have rv'= 



c' 

 -y=, v'= the value of v between r, and r when infinite j but rtan.4' = 



c' . . 



—7^, hence in this spiral ?;'=tan.-^ : it is also evident that if a per- 



pendicular =— 7^=rtan.4' is erected at the centre of the spiral to r 



infinite, and a line is drawn through the extremity of the perpendicu- 

 lar parallel to r infinite, the line thus drawn is an asymptote to the 

 spiral. Ifc'^ is Z. A, but c'^ cosec. =4' > A, c is evidently positive; then 



A — c'' 

 by putting = R^, (5), (6) become (supposing r to decrease) 



c' —dr 1 —rdr 



dv=—r=y. — — i a^=— rnx / — =^' whose mteerals are 

 Vc r-v/r^^+R^ Vc Vr^-j-R^ 



c' , , /RVR"4-r^+R'R\ VW+W^-VW^^ 

 v=- — -xh.lA =^--=^= — -— , t==- 



Rv/c ■ \r^/R2+R''+^R / ^ 



(t?,< commencing when r=R',) or r-= y/C'V^R" +R'^~^v^^)^~R^ • 



This value of r, when substituted in the value of v, gives v in terms 

 of t and known quantities, whence r and v are easily found at any 



^/RM-'R^-R . , 



time which is less than ■;= = the time from the extrem- 



V c 



ity of R' to the centre of the spiral. By taking the integral of 



c' —dr . . c^ 



cZ«=— 7=X — > between r, and r infinite, v'=: ^ ,- Xh.l. 



Vc rv/r^ 4-R=^ R^c 



[ ^_ —1 IL ) , t^'= the value of v between r and r infinite ; hence 



by substituting the value of r, in terms of t, in v', the position of r 

 infinite becomes known at any time ; this curve evidently has an a- 



symptote parallel to r infinite, at a distance =~7^= the perpendicular 



V c 



from the centre of the spiral to the asymptote ; as is evident by 



c' 

 making r infinite in (1), which gives p=—^' If c'^ is /.A, and 



V c 



c'^cosec.^'4'Z A, c in (3), (1) becomes negative, and its sign must 



be changed, also the signs of the terras involving c in (5), (6) must 



A-c'== , c' 



be changed; hence by putting — -— =^2 they become dv=— 7= X 



