334 Central Forces. 
short exposure to the weather, it changes to a dull yellow, and at 
length its surface passes to black. 
Several doubtful substances have been found in the vicinity of 
Amity, of which I have taken no notice in the foregoing remarks, and 
whose character can only be completely elucidated by a better sup- 
ply of specimens than their localities have as yet afforded. 
Arr. XIV.—On Central Forces ; by Prof. THeovore Srrona. 
(Continued from p. 69, of this volume.) 
: A __2a(4): also 
I witt now suppose A= const. and P= =—5 (73) also by 
rT? 
dr 
ane 2 je A 
the general forms of F', [have F= — 7 d (=) ; hence e/?d—, Aa; 
p* P ¥: 
dr 
‘ Be Lakes oe wags 
whose integral is ponte (1), which is the general equation 0 
the curve described by the particle ; the origin of r being at the cen- 
tre of force, and p=the perpendicular from the centre of force to 
the tangent at the extremity of r, also c= the arbitrary const. which 
is easily found in terms of A, ¢’, and the initial values of r, p- t 
V’= the velocity of a particle of matter describing a circle around 
the centre of force at the distance r, V= the velocity in the curve at 
72 
the same distance, then F=>= = or A=V“r?, and —=V’; 
hence (1) is easily changed to V? —V/? =e, (2); let |= the an- 
gle at which r cuts the curve, then p=r sin.) and (1) becomes 
e’*cosec.2.) A ridv? . r2dv 
r? oe to (3). Again, ce’? =—F7a~ gives a= (4); 
dt = the element of the time, dv = the element of the angle descri- 
r vy? e 
bed by r around the centre of force ; also Pp? =a 4ride hene 
and by (1) dv= 5 d (4) becomes 
y ( ) rfer?+A—c? ( %; an ( ) 
+ rd ; 
Tae (6), the sign + being used when #, %, 7 increase, 
The species of 
and the sign — when ¢, v increase and r decreases. 
the initial values 
curves denoted by (1) or (3) depends on A, ¢’ and 
