420 APPENDIX. 



No. II. (p. 58). ANOTHER DEMONSTRATION. 

 TAKING the expression - for the force and making it 



i r 

 = the general form -^-^ where h is constant, we have 



p JLv 



70 



- - = ^ This is a differential equation of the second 

 p 3 . li r 2 



order, and it can have but one general integral, and that can 

 contain only two arbitrary constants. Such an integral has 

 been found ; for it has been shown that any conic section 

 with the focus in the centre of force satisfies the equa 

 tion. The nature of the case shows that any singular solution 

 is out of the question. We may also arrive at this result 



A 2 

 by reversing the process in p. 49. Substituting for R, g x 



dp u, , dp ft, dr i L , 1 2 , 

 -= and -1 =i -p and migrating, ? == -^ 



/&quot;i_l_ cV This is a well known property of all conic sec 

 tions; but to show that no other curves possess it, put 



1 1 du 



- u\ then since = ?r + -=-. 



. r 



, therefore - TT 

 a Q 



M. MA\/U &amp;gt;JL.UV/V^ n 7 



r p 2 d 



V~ 2 a u 



C + -^ u 



2 7^5 . u -f C, and therefore , = d 0. 



