506 Mr. C Gr. Darwin on the Collisions of 



to the line of the bipole are gi 



at right angles 

 equation 



by the- 



I 



sech § 



du 



where k=- = -^ I ^ 

 and 7=—. 



77 



V) Jo 



rfi/r 



\/(7 2 ' 



sin- 



i|r)' 



(9-2) 



+ 



r ) 



The orbits are 

 elliptic integrals, 

 general character 



thus given by equations between two 

 It will be convenient to consider their 

 Consider the left of (9*2), which we 



shall call 1^. As J diminishes from infinity sech f increases 

 from zero and with it I f . This process continues till sech f 

 has reached the value of the least positive root of the quartic 

 under L. These roots are 



1, -1, 



y/tf + P-k \VH-P + £ 



(9'3) 



r 7 



Different cases arise according as the first or third of them 

 is the lesser. If the first, that is if y 2 < 1 — 2&, then sech f can 

 reach the value 1, that is (■ can vanish. This implies that 

 H can go right through between the two halves of a, a state 

 of affairs that we do not require to study, as it obviously will 

 not give a suitable orbit. So we take y 2 > 1 — 2/:. Then 

 sech f can never exceed the value given by the third root or 

 (9'3). The point where it reaches it is a turning point and 

 may be called the apse. After the apse sech If must diminish 



Fie:. 8. 



c 



/..V J 



JB 

 / t 





; 



■secht; 



G^ 



ao-ain, while I* continues to increase ; and when sech f has 

 reached zero L will be double its value at the apse. In fig. 8' 

 there is a sketch of the general character of I, plotted against 



