﻿50 Mr. A. S. Davis on the Probable 



Assuming with Encke that the resistance opposed to a comet's 

 motion varies as the square of its velocity, and inversely as the 

 square of its distance from the sun, the change produced by the 

 resisting medium at any part of the comet's orbit is given by the 

 equation 



da , (1 + ecosw) ¥ 

 -7-= — k- -g* 



au (1 — eeosu) 2 



where e is the excentricity, u the excentric anomaly, and k a 

 constant. 



For very small values of u we have 



da ( ■ eu* W, en* \-f 



- (T^ 1 TTe~ 2 + '"! l l + T=e 2 '"! 



(l_ e )t L 4 1 + e J I 4 1— e J 



(1- 

 Hence 



= __# Z_ ? neglecting terms containing w 2 and ~-r 



(1 — e)a i_ " e 



I 



-~du=—2k- 4 . w, 



_ M flfo (1 — e )T 



if w be very small. The excentric anomaly of the end of the 

 #2 1 _ e 2 



latus rectum is tan -1 -^- = - very nearly, if e does not differ 



a e e 



much from unity. 



Hence the change produced in the semiaxis of the orbit whilst 

 the comet is passing from one end of the latus rectum through 



perihelion to the other end is —2k ^ 3 = — nearly if 



e(l — ef 6 f *' 



e = l — € and e is small. 



The change produced in another orbit will be — 8 ^ ^ • and 



€ 2 



the ratio of these changes is (Z-J \ Let us suppose that the 



two orbits have equal perihelion distances. Then, since a very 

 large portion of the whole change which is produced during one 

 revolution in the semiaxis of the comet's orbit is produced whilst 

 the comet is passing from one end of the latus rectum, through 



