1209 



the well known law of the vis viva. When now we substitute the 

 above found value in pole coordinates for u\ we get: 



/dry c- Fr 



/"dxpy- 



as ?'M — I is ^ c' : r* according to (1). This equation (3) is of great 



importance for our calculations. For the equation of path the following 



d\^ c 



equation follows trom this in connection with — = : 



^ dt r* 



dtb c 



— = ± ^=,=z:,=z ... . . . (4) 



dr i / p,. c* 



Thé upper sign evidently holds for ;• = a to r ^= r^ {A to B), 

 the lower sign for all the points beyond B. ^) 



We must now first determine the constant c. This can take place 



in two different ways. The simplest way is to examine the state in 



the point A, where the path coincides with the tangent AO' . If the 



considered point P lies in the immediate neighbourhood of A, then 



evidently 5m i^if.' — 6) is z=asin6:r in A GAP. Differentiation with 



(ftp a sin 6 



respect to r gives cos{y\^ — 6) -—- =i — — -— , hence because tl; is then 



dr r 



= 180° and r = a : 



'dxp\ ^ t^ 

 dr J ji a 



But from (4) follows, because Pr = in A, and the positive sign 

 holds : 



c^ip\ c 



a 

 From the two expressions follows immediately : 



C = Mj X " *^'* ^• 



This causes (3) to become: 



dr\ f a'sin'ê\ Pr 



hence we find for the point B, where the radius vector r gets the 

 minimum values ?',„, and where therefore dr : dt ^ -. 



d'r c' /(r) 



^) We must remark that by difTerentiation of (3) follows^-;- — — i- = • 



^ dt* r* n 



