277-279] Confocal Coordinates 239 



and as we have already seen that equation (197) is exactly equivalent to 

 Laplace's equation V 2 F = 0, it appears that equation (198) must represent 

 Laplace's equation transformed into curvilinear coordinates. 



In any particular system of curvilinear coordinates the method of pro- 

 cedure is to express h l} h.%, h s in terms of X, p and v, and then try to obtain 

 solutions of equation (198), giving V as a function of X, p and v. 



SPHERICAL POLAR COORDINATES. 



278. The system of surfaces r = cons., = cons., <j> = cons, in spherical 

 polar coordinates gives a system of orthogonal curvilinear coordinates. In 

 these coordinates equation (198) assumes the form 



9 / dv\ , i 8 / . a dv\ , i a 2 F 

 fr(*w) + ^M(* me W + **0w ' 



already obtained in 233, which has been found to lead to the theory 

 of spherical harmonics. 



CONFOCAL COORDINATES. 



279. After spherical polar coordinates, the system of curvilinear coor- 

 dinates which comes next in order of simplicity and importance is that in 

 which the surfaces are confocal ellipsoids and hyperboloids of one and two 

 sheets. This system will now be examined. 



Taking the ellipsoid 



as a standard, the conicoid 



2 i/ 2 z 2 



+ = ..................... (200) 



will be confocal with the standard ellipsoid whatever value 6 may have, and 

 all confocal conicoids are represented in turn by this equation as 6 passes 

 from oo to + x . 



If the values of x, y, z are given, equation (200) is a cubic equation in 6. 

 It can be shewn that the three roots in are all real, so that three confocals 

 pass through any point in space, and it can further be shewn that at every 

 point these three confocals are orthogonal. It can also be shewn that of 

 these confocals one is an ellipsoid, one a hyperboloid of one sheet, and one 

 a hyperboloid of two sheets. 



Let X, //,, v be the three values of 6 which satisfy equation (200) at any 

 point, and let X, //,, v refer respectively to the ellipsoid, hyperboloid of one 

 sheet, and hyperboloid of two sheets. Then X, p, v may be taken to be 



