19] VARIABLES AND FUNCTIONS. 31 



(8) cos * 



(c 2 + X) (c 2 + /*)) V^ 7 (X) F' 

 Now by subtracting from the equation 



a? y* z> 



tf~+\ + ^\ ~ ' 



the equation 



a* 

 a 

 we get 



+x ~ ^TM = Oi 



or 



f 

 (9) (X - A*) 



a? 



(6 2 + X) (6 2 4- /O 



*2 1 



= 0. 



Accordingly, unless X = yu, cos (W X W M ) = and the two normals are 

 at right angles. Similarly for the other pairs of surfaces. Accord- 

 ingly the three surfaces of the confocal system passing through 

 any point cut each other at right angles. 



If we give the values of X, yit, v, we determine completely the 

 ellipsoid and two hyperboloids, and hence the point of intersection 

 x, y } z (and its seven symmetrical points in the other quadrants). 

 Hence we may take X, fju, v for the coordinates of the point, 

 and the family of surfaces forms an orthogonal system. X, /*, v 

 are called the ellipsoidal or elliptic coordinates of the point. 

 We shall proceed to find their parameters in a form not con- 

 taining any coordinates but X, //,, v. We must find the rate of 

 change of X as we go along the normal to the ellipsoid X = const. 



Since we have identically 



a? <L 



-1 = 0, 



