MR TALBOT ON FAGNANI S THEOREM. 



295 



Pig. 13. 



Therefore, calling CN, x^ we have, when the point is at the extremity of the 

 minor axis, 



e : 1 : : HB : BO 



: : HB : CN + PX 

 HP 

 e 



Therefore «=e.t7 + HP 



and HP = a— ex. 



This equation to the ellipse may often be 



useful. 



Theorem A. — If an ellipse and hyper- 

 bola are confocal, the line from the focus 

 to the point of intersection equals the distance between the Vertices. Let S, H, be 

 the foci, P the point of intersection, 

 a, h, the semi-axes of the ellipse ; A, B, 

 those of the hyperbola, and V its Vertex, 



SP + HP=2CA, and SP-HP=2CV 

 •. 2HP=2(CA-CV)=2VA 

 .-. HP = VA = a-A. Fig. 14. 



The distance CH, between the centre and focus is usually expressed by ae, or e 

 times the semi-axis major. But since in theorems concerning two or more con- 

 focal conies, CH is the only invariable line, it is convenient to denote it by unity. 



We will therefore in the sequel suppose CH, or ae—1; and, therefore e=-. It 



must be borne in mind that the condition of confocality gives the following rela- 

 tion between the axes, — 



a2-&2 = i=A2 + B2. 



Theorem B. — The same suppositions being made as in the last theorem, the 

 co-ordinates of the point of intersection have the values 



x = A.a y = 'Bb. 



And CP the central distance of the point of intersection, is equal to VB, which 

 also has the value, — 



For we proved in the last theorem that 



HP=a— A, but by the Lemma ( putting e=-j , 



HP=a- 



X 



- = A, or x = Aa. 

 a 



Therefore, 



It remains to find the value of y 



