Conway — The Dynamics of a Rigid Electron. 179 



where p^"' means d/'^p/cU", can be solved as follows : — Let the real roots of the 

 equation k^f + k^of''^ + . . . = be a^, cin . . , , and let the imaginary roots be 



&1 + fi\/ - 1, &2 + C'i\/ - 1. 



Then if r^, To . . . /3i, ft^ • ■ • a\, a. . . . be constant vectors, and if iii, n. . . . are 

 scalars satisfying 



Ui log Tui = hi ; wni = 2ci, 



n^ log Ta. = Ijo ; TT^io = 2c2, etc., 

 then 



In the present case 



p = y^e'dt + Fai'^i^/Si. 



ISTow if ^^ is small compared with the other coefficients, ai becomes large 

 and is negative, and so y^e^'it is negligible after some time ; the remaining 

 term shows that the electron describes in general a curve which is the 

 projection of a logarithmic spiral. In the second place, suppose that the 

 electron moves in a field of force, the potential of which is p, so that 



- kp' + m'p ^ '^j), mp = V^9 + kp. 



If k is small, we have then approximately 



mn = Vw + — V» = V« V. SpV.p. 



' ^ m. m 



Hence the motion is the same as if there were no radiation and a new 

 force - ^{'p) added, where denotes a linear self-conjugate vector function, 

 the constants of which are functions of p. It is thus easy to see that 

 the motion now is of the type into which a Dissipation Function enters ; 

 in fact, the energy equation is 



^.^r.-.on.,.p-\si>mdt. 



For the case of the law of nature when p = ^l{Tp) 



klf p SpSpp \ 



7. Third Appeoximation. 

 For the third approximation we have 



[^)f>:r(p-nd- 



