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XXXYI. On the Motion of a Particle about a Doublet. 



To the Editors of the Philosophical Magazine. 

 Gentlemen, — 



IN a recent paper* Sir J.> J. Thomson investigates the 

 motion of an electron about a doublet,- I have recently 

 been examining this same motion, and, as I have arrived at 

 very different conclusions from those of Sir J. J. Thomson, 

 a brief statement of my results may not be out of place. 



The equations of motion of this particular problem admit 

 of complete integration, providing, I think, the only instance 

 in particle dynamics of a soluble problem in which the orbit 

 is not confined to one plane. 



As stated by Sir J. J. Thomson, the equations of motion 

 are 



& r • 2/)i2 fr 2M<?cos0 



dt* r m r 3 v J 



d , * «, . „ a o Me sin /os 



J t (r*6)-r>™eco S ep-=- -- -s , . • (2) 



J(rW0£)=O, (3) 



and these hate the obvious first integrals of momentum and 

 energy, 



r 2 sin 2 0</> = 77, (4) 



l(r« + r*^ + r*sin 2 0*)=E + — ^i?. . . (5) 



- v r j m r 2 ■ y 



Equations (2) and (3) give 



f zh\ d 1 2 h\ 2 d f 9 As ^ e • /, ™ 2 cos # 



7 dv v ; dt y ' m sir 6 



leading to the integral 



^^•".^JL + t . . (6) 

 2 m r 2 2r 2 mi~0 r 2 ' 



in which C is a constant of integration. From this and (5), 

 of which the integral is 



^' 2 = E-^, (7) 



* " On the Theory of Radiation," Phil. Mag. xx. p. 244. 



* 



