1719] VARIABLES AND FUNCTIONS. 27 



The parameter of a function f(z, p, CD) is the resultant of the 

 partial parameters 



V . 8 / - 18 / 



~ 



19. Ellipsoidal Coordinates. The equation of a central 

 quadric surface referred to its principal axes is 



where a 1} a 2 , a 3 may be positive or negative. If they are all 

 negative, the surface is imaginary. 



1. Suppose one is negative, say 



while ! = a 2 , a 2 = & 2 - 



Let a > b > c. 



The equation is^+^-^ = l. The surface is cut by the XY- 



plane in the ellipse ^ +^ = 1, whose semi-axes are a and 6, and 

 o, J cr 



whose foci are at a distance from the center 



The section by the ^Z-plane is the hyperbola 



___ 1 

 a 2 c 2 ' 



with semi-axes a and c, and foci at distance Va 2 + C 2 = Vaj a s on 

 the JT-axis. 



The section by the F^-plane is the hyperbola 



with semi-axes 6 and c and foci at a distance \/6 2 + c 2 = Va 2 3 on 

 the F-axis. 



The surface is an hyperboloid of one sheet. 



