y (33) 



Classification of Electromagnetic Fields. 143 



and magnetic vectors may be expressed in the forms 



x M dr L ioL + mP + ny— 1 Z a + m o/3 + n 7 — 1 J ' 



_ i d r v/3 — fiy v /3 — /yy 1 



~MrfTLXa + /A/8 + V7— 1 X a + ^ + ^07 — 1]' 



H i — r 0)i — ym _ @n — ym n 



i d f X — a X-o — a -j 



M <^T [_A,a -+ /jb/3 + vy — l X a -+- //, /3 -f v y — lj 



With the aid of these expressions we can show that the 

 lines of electric and magnetic force on a sphere whose 

 centre is at (f, 77, f, r) are such that when the sphere is 

 inverted into a plane the electric lines of force are repre- 

 sented by the equipotential lines due to two doublets, and 

 the magnetic lines of force by the corresponding stream 

 lines, the flow being in two dimensions. The bullets are 

 thus magnetic doublets. 



§4. A very general type of electromagnetic field in which 

 the electric and magnetic f orces at #, y, z, t are perpendicular 

 to the radius from f , tj, f, r may be obtained as follows : — 



Let w, iv , <t be defined by equations (20) and write 



3fe.v)' 



y (34) 



H -MM H -Mill H - 9 ^> T ) 1 

 a * ~ B(y, zy **' B( 2 , «) ' M * - 3(.t-, y) ; J 



E _w,t) _ _b(^t) _ a(g,r) 



Then'-if 



IV 



we have also 



*£■ ')-'&-) 



)• (35) 



H __B0n) H _. B(v,t) „_ aOvr) 



u * 90,0' *~ B(y,0' *~~d(J70 ; 



E *~ B(y,,)' E *~ 5(^' ^~ 5fcy)J 



and it is evident from these two sets of relations that 

 Maxwell's equations (1) are satisfied. 



