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. 



Art. XIV. — On Central Forces ; by Prof. Theodore Strong. 



(Continued from p. 69, of this volume.) 



A A / 1 



A A / 1 \ 



I WILL now suppose A= const, and F-- — = — ^^l"]: also by 



c'' ( 1\ 1 1 



the general forms of F, I have F= — -^d{ — ]; hence c"^d—=Ad~, 



dr 



c'= A ^ . . . , 



whose integral is ~~2=^~:^-\-C', (1), which is the general equation of 



P" ^ 

 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, c', and the initial values of r, p. Let 

 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 



A V- c'2 



ihe same distance, thenF = -5 = — or k=-Y'-r-, and —^ =V2 ; 



ihence (1) is easily changed to Y^ —Y'^—c, (2); let 4^= the an- 



■s;le at which?- cuts the curve, then p=rsin.-^ and (1) becomes 



.c'='cosec.2 4. A , . , . r'^dv'^ . r-dv 

 -. = -^ + c, (3). Agam, c''-=-^^ gives dt=^-^, (4), 



dt = the element of the time, dv = the element of the angle descri- 



r^dv^ 

 .bed by r around the centre of force ; also p"= 7 o , . ; .' hence 



— I— c^r 

 and by (1) do = r=== (5), and (4) becomes dt = 



, (6), the sign + being used when t, v, r increase, 

 V cr- +A — c'^ 



and the sign — when t, v increase and r decreases. The species of 

 curves denoted by (1) or (3) depends on A, c' and the initial values 



