428 Prof. Rowland on the Propagation of 



where 



B? = a? 2 +y 2 +* a and r 2 = x' 2 +y' 2 + z /2 . 



Hence one can write 



__ .p xx r + yy ' + zz f r 2 



_ #,2/ + yy f + 2/ r 2 

 ? = ~~ c 1 +C 2R' 



When E is great and r is small, the last term will vanish and 

 leave a very convenient form for many cases. 



But it is not necessary to always perform the calculation at 

 a finite distance ; for we have the following theorem, which I 

 believe to be new. 



Theorem. Supposing the sources of light to be continuous, 

 and knowing the light over a sphere at an infinite distance, it 

 can be determined for all space by the following process : — 



Let F, G, and H be the components of any of the vectors, 

 such as the electric displacement &c, and suppose them known 

 over the infinite sphere. Express them in terms of series of 

 harmonics, thus 



F=^ {E Y / + E 1 Y 1 ' + E 2 Y 2 ' + &c.}, 



r 



G=^ {E Y " + E 1 Y/' + E 2 Y 2 " + &c.}, 



r 



n pc(o-vo 



H= ^f {E Y "' + E 1 Y/" + E 2 Y/"+ &c.} 



P 



At any other point of space the equations must satisfy the 

 equations of light motion and also the equation of continuity. 

 Referring to the equations of p. 418, we see that if we multiply 

 each term by the corresponding quantity C n , the equations of 

 light will be satisfied, and if the surface harmonics are of 

 the form 



y/_- n njT ^V-(*+l) dV-(n+l) \ 



Xn ~ P l y dz Z dy f> 



ln ~ p V dx ~ X ~~~dz )> 



ln - p r dy y dx j> 



the equation of continuity is also satisfied. Hence the com- 

 ponents of the vector at any point of space are 



