151, 152] GREEN'S FUNCTION. 293 



, so that T? T' P . Accordingly we have for the two points 

 P and F, 



(10) GV = IV + - 



In order to show the dependence of the function G on the co- 

 ordinates of its pole P let us write it 



(! !) Q(x,y,*) = 9 O, y, z> a, b, c), 



and G' (x, y,z) = g (x, y, z, a', &', c'). 



Then by the above theorem 



G(a',b',c')=G'(a,b,c), 

 (12) g (of, b', c', a, b,c) = g (a, b, c, of, 6', c'), 



or Green's function is a symmetric function of its variables a, b, c 

 and a', 6', c'. 



152. Examples of Green's Function. Plane. Let us seek 

 Green's function for all that portion of space lying on one side of 

 a given plane. Let A be the given pole, at a distance a from the 



r. 



A/ *rl B 



Fio. 62 a. 



plane, on the left, and let B be its geometrical image in the plane. 

 Let the distances of any point at the left of the plane from A and 

 B be r and r respectively. Now for every point at the left of the 



plane the function -, is harmonic, and for points on the plane, 



where r = r, it assumes the value . It is therefore the function 

 F of the preceding article. We have then 



- 



"r 7' 



dG _ cos (rii r) cos fa r') _ 2 cos 

 dn~i~ ~r*~ r' 2 r 2 ' 



where 6 is the acute angle included between the radius r and the 

 normal to the plane. Consequently, the equation 



