144 Dr. H. Bateman on the 



To verify these identities we have to show that if 



3fe _ i 'd(J 1 r) 3Q, t) _ . 3Q, t) 30, t) _ .30, t) ,_ 



3(^)-*30M)' B(*,«) 3(?/,0' 30^) §(m)- l j 



Now we find at once that 



(&+<£?■+&'-(* 



3f 2 



~dtj 



3^3t ~ds 3t 3s3j_3*3t 

 3^ + 3y 37/ + 3i 3* ~ 3* 5F' 



©** (*)*-©■=©' 



3f 30, t) 3s 30, t) 3f 30^t) _ 

 ' ' 3^'30, + V3(j/, + 3-*30, t; " ' 

 3t 30% t) 3 j 30, t) 3t 3Q, t) _ 



Hence it will be sufficient to verify the first of the 

 equations (36). To do this we must show that 



3t 3t -x 3 T • 3t\ / 3t 3t . 7 3t . 3t 



/ 3t 3t ., 3t • ot\ ( 



-by l,x -bt ^3^r~V m 3^ n 3y ' Z 3^ ?i V 



If now we use the values (24) this relation is easily seen. 

 to be satisfied in virtue of (6) . We may infer, then, that it is 

 satisfied in the general case. 



In the electromagnetic field (34) the electric and magnetic 

 forces at (#, y, z, t) are at right angles to the radius from 

 (?» Vi Kt T ) 5 tne .Y are a ^ so a ^ right angles to one another and 

 equal in magnitude. The moving point (f, 77, f, r) again 

 has the character of a gun which tires out bullets which 

 move with the speed of light and are singularities of the 

 electromagnetic field. 



The lines of electric and magnetic force on a sphere whose 

 centre is the point (f, 77, f, t) are easily drawn. It follows 

 at once from (34) and (35) that the magnetic lines of force 

 are given by = constant and the electric lines of force by 



v= constant. By choosing the function /( -, t ) in a suitable 



way we can make the distribution of the lines of force satisfy 

 certain prescribed conditions. 



