Conway — The Dynamics of a Rigid Electron. 175. 



length and keep constant their mutual inclinations so that their motion 

 depends on an angular velocity w, then 



^ 0(r) = (/< (r) + Sfi/S'/Br + Sa.S'jSr 



(jt _ - 



= 0(r) + 2Fa;a>S'j3r + So>Sw/3r 

 = 0(r) + Fo^^(r) + 0(Ftw). 



In particular 



— 7 (p (w) = f/)&) + F't.K/)W. 



at 



Applying these results to the expressions for A and fx obtained in the 

 last paragraph, we get 



A= -g^(^0:(O + fiHJ, 



dl 



d 



dt 



so that the equations of motion are 



iu = -g^ i Vt(<1>,(t) + f,(to)) + <^/(r} + rp,{uy)\ 



^ I Fr(^:(r) + 0.(co)) + -^^(r) + <PM\ = /'• 



To get the activity we multiply X" by f and the moment about the extremity 



of T, i.e. u" - FrA", by w. It takes the form — - where 



dt 



E =^ {S'(J)i(t) + 2ST(p2{h)) + Sw<j>;(jj], 



so that JE represents the Kinetic Energy of the system. The forms of U and 

 of the two equations of motion are identical with those we meet with in the 

 irrotational acyclic motion of a body through a liquid.* These equations have 

 received much attention from various mathematicians especially for the case 

 of no forces. It is therefore unnecessary to mention more than one case. If 

 there are no' forces and no rotation, we get 



Fr0i(r)+</3(r)=O. 



From which r = and 0i(r) || r. 



Hence f is the direction of one of the axes of the function ^i. These are 

 rectangular, for it may be seen that ^i is self-conjugate. We thus gee 

 0ir = gr, where g is one of the roots of the symbolical cubic of ^i, showing 



* Cf . Lamb's "Hydrodynamics," chap. vi. Joly's "Manual of Quaternions," p. 241. 



