Some Orbits of an Electron. 205 



decrease also. When p = a cot ft, tan jjl= sin ft, and % = 0. 

 Then 



a6' = 



^{l + ^iocot//, — w 2 cot 2 yu, tan 2 ^} ' 



and in te oration gives the orbit 



20 cot fi tan ^ sin ^= cos% — cos (#cot //,tan % + %)'• 



This is a curve modified from one limb of an hyperbola 

 with the nucleus in the focus, the modification consisting in 

 increasing every element of vectorial angle in the ratio 



/ a 2 c 2 

 tan yu, cot % : 1, that is the ratio 1 : \/ 1 - 2 — 



V2- 



f V 2 " 

 The second asymptote occurs when 6~2{ir— %) tan /a cot ^. 



a 



When - cot/3 is small, that is either when p is large or 



the velocity very high, then the orbit is nearly a true hyper- 

 bola. For smaller p or smaller velocity the curve tends to 



wind round the origin. When — cot/3 approaches unity the 



p 



electron will describe several turns before it can escape. 

 The time integration is perfectly simple. It gives 



[sec 2 / v 

 _. ^L 

 t t + tan ' 



tan (^ () cot fi tan x). 



X 



+ 2 cos 2 ft cot X log T+ ^°* 1 



II. p = a cot ft. 



In this critical case we have 



irk dw 



av = 



y/{l + 'Awcosec ft} ' 

 from which 



w = d{l + ^Ocosecft). 



Since w—pjr the curve ultimately approaches the origin 

 like the spiral r = A/0 v 2 . 

 The time is given by 



Toil -» cosec B l~- i 1 . cos^ fl + 2sin/3 -| 

 let] peoteop^ 0-0 + 2sin/3 + sin/3 ll>8 J' 



The time taken to arrive at the origin from any finite 



