46 Central Forces, 



The notice of his obsequies arriving only at the same time with the 

 news of his death, did not allow his fellow members to perform the 

 accustomed honors at his tomb. But if such honors, if any efforts to 

 extend renown and render it durable, were ever superfluous, it would 

 be for the man who, by the happy choice of the subjects of his labors, 

 had richly earned the esteem of the learned, and the gratitude of 

 the unfortunate. 



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



(Continued from Vol. XVII. p. 73, of this Journal.) 



I HAVE proved at pages 72, 73, of the Journal for October, 1829, 

 that the value of F in the case of a particle of matter describing a 



conic section, about a centre of force at the focus, is —r~, ' hence 



A ^ 

 conversely, supposing the same notation, and that F=-:^' (A=const.) 



the nature of the curve described is easily found. For assume 

 --r=A, thenp'=^- is found; .•.F=^,= -^-f?|^— ,J ; but 



dr 



C / 1 \ 



the general form of F given at page 72 is F= — ~2~^f ^2 gj^ 21 ) 5 



dr 



hence, by comparing these values of F, I have d i —^- — ^ j = 



/2 \ 1 2 1,1, 



di — ^/ 1 3 or by integration -^7—^ — rT~ "~~/ ~ — / (1)3 ( — — "/=the ar- 

 \rp'J ' -^ o r- sm. ^4' ^i^ W ^P 



bitrary constant.) (1) is the general equation of the conic sections, 

 p'=the semi-parameter, a=the semi-transverse axis, r=the dis- 

 tance of any point of the curve from the focus, and r sin. 4^= the 

 perpendicular from the focus to the tangent at that point, and it is 



1 



evident that the centre of force is at the focus. By (1) I have 73= 



2 1 



Td'~ r'^ sin ^%L ^^ ' '^^ ^ ^"^ '^' ^^"°^^ ^^ given values of r and 4 



