240 Methods for the Solution of Special Problems [OH. vm 



orthogonal curvilinear coordinates, the families of surfaces X = cons., //, = cons., 

 v = cons., being respectively the system of ellipsoids, hyperboloids of one 

 sheet, and hyperboloids of two sheets, which are confocal with the standard 

 ellipsoid (199). 



280. The first problem, as already explained, is to find the quantities 

 which have been denoted in 2*77 by h ly h 2 , h 3 . As a step towards this, we 

 begin by expressing x, y, z as functions of the curvilinear coordinates X, p, v. 



The expression 



is clearly a rational integral function of of degree 3, the coefficient of 6 s 

 being 1. It vanishes when is equal to X, p or z/, these being the curvi- 

 linear coordinates of the point x, y, z. Hence the expression must be equal, 

 identically, to 



Putting 6 a? in the identity obtained in this way, we get the relation 



2 (6 2 - a 2 ) (c 2 - a 2 ) = (a 2 + X) (a 2 + /*) (a 2 + v\ 

 so that x, y, z are given as functions of X, //,, v by the relations 



(a 2 + X)(a 2 + / .)(a 2 +.) 

 2 _ 22 - 2 



281. To examine changes as we move along the normal to the surface 

 X = cons., we must keep /JL and v constant. Thus we have, on logarithmic 

 differentiation of equation (201), 



dx _ d\ 



' ~x ~ a 2 + X ' 



and there are of course similar equations giving dy and dz. Thus for the 

 length ds of an element of the normal to X = constant, we have 



,7 >c (a 2 +X)(6 2 -a 2 )(c 2 -a 2 ) 



The quantity ds is, however, identical with the quantity called -= - in 



"i 



277, so that we have 



2 2 2 



