108 Dr. H. Bateman 



on an 



of the aether, and so it may be worth while to present the 

 analysis in a compact form. 



2, In a former paper * an attempt was made to describe 

 the motion of the lines of force of an electric pole by means 

 ot a succession of infinitesimal transformations which trans- 

 form light-particles into light-particles. If for simplicity 

 we consider only transformations which correspond to rigid 

 body -displacements in Minkowski's four-dimensional space, 

 the path curves or trajectories of the transformations may 

 be written in the form 



du = dTl v \i) + c{t-T)n-{x-^f+U-i)li'], i 



dz = dT[?(T) + c(t-T)r-(x-Z) 9 + (y- v \f], ! 



cdt=dT[c + (a:-Zyp + (y- V )q + U-Z)r-}, J 



where £, rj, f,/, g, h, p, q, r are functions of t. The infini- 

 tesimal transformation associated with a particular value of t 

 is supposed to be applied to space-time points [x 3 z/, z, t) 

 which satisfy the relation 



and also to certain space-time points specified by equations 

 of type 



*=Fi(X, T, Z, t), s=F 3 (X, Y, Z, t), . 



y=F 2 (X,Y, Z,r), « = F„(X, Y. Z, t), } ' U 



which may be regarded as solutions of the differential equa- 

 tions (1), X, Y, and Z being constants of integration or 

 invariants for the sequence of transformations. 



The equations (3) may be supposed to give the co-ordinated 

 motions of the different points of an electron of which 

 (£? y> ?? T ) is a particular point which we shall call the 

 focus. The lines of force of the electron are supposed to be 

 generated by light-particles fired out from the different 

 positions of this focus S, the direction of projection of the 

 light-particles associated with one line of force varying in a 

 manner indicated by the succession of infinitesimal transfor- 

 mations. In fact, if (/, m, n) are the direction cosines of the 

 line of projection of the light-particles emitted at time t we 

 have 



-j- — p—mh + ng — l{lp + mq + nr). . . . (4) 



Proc. London Math. Soc. (2) vol. xviii. p. 95 (1919). 



