238 Methods for the Solution of Special Problems [OH. vin 



GENERAL THEORY OF CURVILINEAR COORDINATES. 



277. Let us write 



c/> (#, y, z) = X, 



"^ (x, y, z} = p, 



X (x, y, z} = v, 



where </>, ty, % denote any functions of x, y, z. Then we may suppose a point 

 in space specified by the values of X, p, v at the point, i.e. by a knowledge of 

 those members of the three families of surfaces 



(x, y, z) = cons. ; -fy (x, y, z) = cons. ; ^ (#, y, z) = cons. 

 which pass through it. 



The values of X, p, v are called "curvilinear coordinates" of the point. 

 A great simplification is introduced into the analysis connected with 

 curvilinear coordinates, if the three families of surfaces are chosen in such 

 a way that they cut orthogonally at every point. In what follows we shall 

 suppose this to be the case the coordinates will be " orthogonal curvilinear 

 coordinates." 



The points X, p, v and X + d\, /z, v will be adjacent points, and the 

 distance between them will be equal to d\ multiplied by a function of 



X, p, and v let us assume it equal to -j- . Similarly, let the distance 



hi 



from X, p, v to X, p -f dp, v be -j- , and let the distance from X, p, v to 



*J 



7 i_ dv 



X, p, v + civ be j- . 

 ri 3 



Then the distance ds from X, p, v to X + d\ /JL + dp, v + dv will be 

 given by 



this being the diagonal of a rectangular parallelepiped of edges 



d\ dp , dv 

 -T- , -j and j- . 

 r^ h 2 fi 3 



Laplace's equation in curvilinear coordinates is obtained most readily by 

 applying Gauss' Theorem to the small rectangular parallelepiped of which 

 the edges are the eight points 



X + ^d\ p ^dp, v \dv. 



In this way we obtain the relation 



-dS = (197) 



in the form 



