298 MR TALBOT ON FAGNANl's THEOREM. 



becomes BH (see fig. 14). Our theorem asserts, that in this case CA = BH, the 

 truth of which is otherwise manifest. 



Again, let CA, CB, and CX, CY, be the semi-axes of 

 the two confocal ellipses, but let the confocal hyper- 

 boles be reduced as before to the straight lines CBY 

 (produced to infinity), and AX (produced to infinity), 

 the intersections of the first hyperbola with the ellipses 

 "aTx will be B and Y, those of the second A and X. The 



Fig- 16- diagonals will become the lines BX and AY ; and our 



theorem asserts that in this case BX = AY, which may be proved independently 

 as follows : — 



Let the semi-axes of the smaller ellipse be a, b, and of the larger one a, (3. 

 Since they are confocal, a^-b^ = a2-/3^ and therefore 



a^+^'-^a' + b-. 



But a^+^^ = AY2, and a^ + 6^ = BX^. 



Therefore AY=BX. 



This theorem may be thus enunciated :— 



" If the alternate vertices of two confocal ellipses are joined, the lines joining 

 them are equal." 



The co-ordinates of (x, y) the point of intersection of a confocal ellipse and 

 hyperbola, may also, if preferred, be readily deduced from first principles, as 

 follows : — 



Given the confocal ellipse and hyperbola whose equations are -2 + ^ = 1 and 



xa— g2 = 5» with the condition that a- — &^ = A- + B^ = 1 to find the values of x and i/ 

 the co-ordinates at the point of intersection ? 



&2 B^ 



The equations give respectively y^ = b-- -jx^ and y^=-^. x'^-B^. Equating 



/Wa^ + 6^ A*\ 



these values of ^^ we find 'B^ + b^=x- i — ^2^5 — ) • Now, substituting l-A^ for 



B^, and a^-1 for b\ we find that B'^ + P=a^-A^, which is also equal to B^a^ + b^A', 

 and therefore may be omitted on both sides of the equation, which reduces itself 



to 1 = Tz-i whence cc=Aa. 



Additional Note. — In the second page of this Memoir it is said, that the total 

 deviation tends to increase without limit. To this it may be objected, that the 

 successive deviations may possibly form a diminishing series, having a finite sum. 

 But it can be easily shown that two successive deviations, when they are small, 

 are (very nearly) equal to each other. Moreover, after diminishing to a certain 

 extent, the deviations increase again, having one maximum and one minimum 

 value in the course of one entire revolution round the circle. 



