155, 156] ELECTRICAL IMAGES. 



If x, y, z are given, we know #, y, z and the position of M, so 

 that we may consider x', y', z given by the above equations, as 

 curvilinear coordinates of the point M, disregarding for the present 

 their relation as rectangular coordinates of the inverse point M '. 

 Forming their differential parameters, 



a# 2 (i _ 2^ 



dx ~ I/ 2 I 4 

 (4) ^r = - 2 ^ 2 ir> 



to 



It is easily seen that the surfaces, x' const., y' const., 

 / = const., cut each other orthogonally, for example the cosine of 

 the angle between the normals of x' = const, and y' = const, is 

 proportional to 



ty Md dtfty_ p4 [4 (x*y + xy* + xyz*) 4ney] _ 

 x dx + dy dy ^ 3z dz " \ I s ' I* J = 



We have then by 87, (5) 



a /iar\ a 



and performing the differentiations 





^ M ' e 



Now we have 



ic' 3iE' 



, , 



' '' + ' ' " 



