Céntral Forces. 65 
Arr. VIl.—On Central Forces ; by Prof. THruopore Srrone. 
(Continued from Vol. XIX. p. 49,) 
By (5) given at p. 48, Vol. XIX, 7 eee > (1)5 or; since p’ 
te a(1—e*) ae p’*do 
=a(1—e*), eas Or bos. are (ae hence c/dt=r? = (Teco. v)2 
=e P 2dv 
(3). Put ae and VJ E=n, then by (3) mins ee Sos, v)? 
(4); nt=the mean eae v= the true do.; which are here 
ae(1 — e?)sin. vdv 
counted from the perihelion. By (2) dr= (1-Fec0s. 2)? and 
a-7r\? 
fies x A / a2 — (s—* (1—e? 2dp 
sin. y= — ; hence ndt= : 
er (1+ e cos, v) 
rdr 
ene SSS eee 
a \/ 2 is Taye 
a 
(5); then ndt=(1—ecos. 9) dp or by integration nt=9—esin.o 
3 9= the eccentric anomaly, which is supposed to be reckoned 
“=e cos. g or r=a(1—e cos. 9) 
—e? 
from the perihelion: by comparing (2) and (5) L-ecos.0~ | 
1+ 
~ € cos. 9; hence tan. 5 eT Sar g (7). Ife=1, the conic 
section is a parabola; and since 1+cos. v=2 c08.25=———; 
: 1 +tan.*5 - 
Hf rd 
(1) becomes r= P == — 5) (8) ; and (3) becomes 
v 
4 cos. *5 
v 
edt ( 1+tan.?5 + \dtan.3 tan. 5? oF by integration ¢’t= a le! + 
tan. ak 
3 put ct=2A, then 2 (3 tan.5 “gt tan. 23) =3A(9). Put 
a 
4 =4 and assume R cos. =, R cos. (vd) =5 (10); hence R= 
Vor. XX.—No. 1. 9 
