Central Forces. 337 
pote dr oO verde 
r/ Re —. Mise + Xa 
isnever 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, 
pes) the value of r in these equations 
supposing t, 7 to commence when r=R, I have v=——=xA.l. 
RYc 
R+VR? —r? (7),¢= =5=xV RF —r? or r=v R? — ct? (8), o= 
(2: 
c R+itv 
he Ae ae uae r, v are easily found by (8), (9) at any 
Ryc WR? — ct? 
time which is Jess than amihe time from oa 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. 
bifor R= his CV, v= VCP, r=CT=CP, and ALBERS 
m= 
is as the hyperbolic sector VCR. 
Ife’? is >A, cis positive, as is evident by (1 ) or (3); or if the 
central force should be repulsive c will be positive, for A is negative 
in this case ; and the signs of ai terms involving A in (1), (3) must 
ies CF (the sign — being used 
be changed: hence by pala 
when the force is ee and “the sign + when it is oe 
(5), (6) become i Re =) dt= Tae ———- 
least value which r can have in ine equations is evidently 2% and 
it is evident that r cuts the curve at right angles when it=R; hence, 
Stpposing that v, ¢ commence when r=R, by taking the integrals, I 
¢ r2 —R? Ja aR?. 
= =-—),t= rr=Vv ct?+R 
Rove xare( sec. A)? i= Vrain ; 
—— Xare (sec.= (eR) ) , whence 7, v are easily found 
Rye 
at any time; also r-=R xsec. — : °); , (10). 
(10) agrees with Newton’s construction; Prin. prop. 41, cor. 3, 
Ruy 
have p= 
ee his fig. 5. for R= his CV, o= ang. VCP, r= 
Vou. XXI.—No. 2. 43 
