:278.] PRINCIPAL AXES. I $i 



where X is an arbitrary parameter. For, whatever value may be assigned 

 in this equation to A, the distance of the centre O from either focus will 

 always be V 2 4- X ( 2 -+- X) = Vfl 2 & ; it is therefore constant. 



276. The individual curves of the whole system of confocal conies 

 represented by (19) are obtained by giving to X any particular value 

 between oo and -f oo ; thus we may speak of the conic X of the 

 system. 



For X = o we have the so-called fundamental conic x 2 /a 2 + y~/t>* = i ; 

 this is an ellipse. To fix the ideas let us assume a~>b. For all values 

 of X > 2 , i.e. as long as 3 2 < X < oo, the conies (19) are ellipses, 

 beginning with the rectilinear segment SS' (which may be regarded as 

 a degenerated ellipse X = ft whose minor axis is o), expanding gradu- 

 .ally, passing through the fundamental ellipse X = o, and finally verging 

 into a circle of infinite radius for X = oo. 



It is thus geometrically evident that through every point in the plane 

 will pass one, and only one, of these ellipses. 



277. Let us next consider what the equation (19) represents when X 

 is algebraically less than b z . The values of X that are < a 2 give 

 imaginary curves, and are of no importance for our purpose. But as 

 long as a 2 < X < /5 2 , the curves are hyperbolas. The curve X = ft 

 may now be regarded as a degenerated hyperbola collapsed into the 

 two rays issuing in opposite directions from S and S' along the line SS'. 

 The degenerated ellipse together with this degenerated hyperbola thus 

 represents the whole axis of x. 



As X decreases, the hyperbola expands, and finally, for X = a 2 , verges 

 into the axis of y t which may be regarded as another degenerated 

 hyperbola. 



The system of confocal hyperbolas is thus seen to cover likewise the 

 whole plane so that one, and only one, hyperbola of the system passes 

 through every point of the plane. 



278. The fact that every point of the plane has one ellipse and one 

 hyperbola of the confocal system (19) passing through it allows us to 

 regard the two values of the parameter X that determine these two 

 curves as co-ordinates of the point ; they are called elliptic co-ordinates. 

 If x, y be the rectangular Cartesian co-ordinates of the point, its 

 elliptic co-ordinates X 1? X 2 are found as the roots of the equation (19) 

 which is quadratic in X. Conversely, to transform from elliptic to 



