19] VARIABLES AND FUNCTIONS. 29 



Its equation is 



a; 2 2 i 2 



r ~~ ' 



where p is to be determined. Clearing of fractions, the above is 

 (2) (a 2 + p) (6 2 + p) (c 2 + p) - tf (V + p) (c 2 + p) 



- f (c 2 + p) (a 2 + p) - z* (a 2 + p) (6 2 + p) =f(p) = 0, 



a cubic in p. Putting successively p equal to x , c 2 , 6 2 , a 2 , 

 and observing signs of /(/?), 



= + oo + 



- *? (6 2 - a 2 ) (c 2 - a 2 ) 



The changes of sign of f(p) show that there are three real roots. 

 Call these X, //,, v in order of magnitude. X lies in the interval 

 X > c 2 necessary in order that the surface may be an ellipsoid, p 

 in the interval c 2 > /JL > 6 2 that it may be an hyperboloid of one 

 sheet, and v in the interval 6 2 > v > a 2 that it may be an hyper- 

 boloid of two sheets. There pass therefore through every point in 

 space one surface of each of the three kinds. If we call 



(3) 



the equation F=0 defines X as a function of x, y, z, and there- 

 fore as a point-function. The normal to the surface X = const, has 

 direction cosines proportional to 



ax ax ax 



a# ' dy ' dz 

 Now since identically F = 0, 



ox oy oz ax 



and we have 



ax /d\\ dx 



ax 



