( 408 ) 



perfectly reflecting on the outside. If, under these circumstances, the 

 surface <y surrounds that envelop, we may put in every point of it 

 a =z 0, (£' = 0, Jp = 0, Sp' = 0. 



b. If the en\'elop is made of a perfectly conducting material, 

 both the electric force ^ and the force (£' will be normally directed 

 in every point of its inner surface. Consequently, if the latter is 

 chosen for the surface o, we shall have 



[i:..ip'l„ = and [^'.Xp]„ =0. 



c. Finally we may conceive a system lying in a finite part of 

 space and surrounded by aether, into which it emits rays travelling 

 outwards to infinite distance. Taking in this case for a a sphere of 

 infinite radius, we shall show that for each element da the factor 

 by which it is multiplied in the equation (25) vanishes. The direction 

 of the axes of coordinates being indeterminate, it will suffice to 

 prove this proposition for the point F in which the sphere is cut 

 by a line drawn from the centre in the direction of the axis of ^. 



Now, if we confine ourselves to those parts of ^, -f?, ^' and ^' 

 which are inversely proportional to the first power of OF, as may 

 ol)viously be done, we may consider the state of things near the 

 point F as a propagation of vibrations in the direction OF, the 

 electric and magnetic force being perpendicular to that direction and 

 to each other. Denoting by a and h, a and b' certain complex 

 quantities, we may write 



0:,. = 0, (2y = ae'"', ^z = ie'"^ 



Jp^ =0,S;)y= — he^'U Xp. = ag"", 

 (E'^ = 0, d'y = aV"', (I'z = b'e"^f, 



Xp'.^ z=^ 0, Sp'y = — &'^"", Xp', = a'g»", 

 and we have at the point F, since in it the normal to the spherical 

 surface is parallel to the axis of x, 



These considerations show that in many cases the equation (25) 

 reduces to 



C\{^\ . d) - (j^'e . t)\ ds =ƒ[((?, . d') - {-% . "by^ds 



(26) 



^ 7. It is particuhirly interesting to examine the effects produced 

 by an electromotixe or a magnetomotive force which is confined 

 to an infinitely small space S. Let F be any point of this region, 

 a a real vector having everywhere the same direction h and the 

 same magnitude |a|, and let us apply in all points of S an electro- 

 motive force a e'"'. Then we shall say that there is an "electromotive 



