206 Mr. C. G. Darwin on 



distance is finite. The time taken for the circuit from 6=.2irr 

 to = 27r(r + l) is approximately 



, , \— a cos ft 



At any point of the orbit the mass of the electron is given 



by ml sec ft + -w J. This will increase indefinitely, but not 



with any extreme rapidity. Thus several rounds may be 

 performed before it is necessary to remember that the mass 

 of the nucleus is only finite. 



III. p< a cot ft. 



In this case we use as before the subsidiary angle fi where 



cot 11= -cot ft cosec ft, 

 V 



but we now take as second subsidiary the quantity co where 

 tanh 2 co = sin 2 ft — tan 2 jju, 



so that 



a 2 



cot 2 a tanh 2 co = — , cot 2 ft — 1 . 

 p l 



When p = a cot ft, tan//,= sin/3 and co = 0. 



(X 



As - cot/3 is increased, fi decreases and co grows. When 



P 

 p = or ft = 0, we have /j, = and tanh«= sin ft, so that co 

 has some finite value. 



Then 7/1 dw 



du = 



*y{l + 2w cot jA+w 2 cot 2 fju tanh 2 o>}' 

 and integration gives 



w cot fi tanh co sinh co = cosh (6 cot /m tanh co-{-co)~ cosh co. 



This curve ultimately approaches the origin like an equi- 

 angular spiral of angle 



eot "V{?^" _1 }- 



It reaches the origin in a finite time, and the time in any 

 position is given by 



[ct~\ = A p cosec ft cot a tanh co \ — ~ — ; — : — 



L J 2r M r L t T + tanhw 



, o 9 a -i i t + tanh co~] 

 + 2 cos- /3 coth co log 



T J t= tanh (^0 cot /x tanh w). 



The mass of the ft particle increases exponentially, but 



