MR TALBOT ON FAGNANI S THEOREM. 



293 



Additional Remarks. 



I will here add two or three other theorems, which have suggested themselves 

 in the course of this inquiry. 



If BCA is an ellipse, and P is Fagnani's point, 

 and the tangent OPT is drawn terminated by the 

 axes produced, then OP = CA, and PT=CB. 



Demonstration. — Let CA=«, CB=Z), CN=^, 



2 2 



PN=?/. The equation is ^+|2 = 1- At Fagnani's 



point we have, 



a? 



,3 53 



"- — 7, and 2/^ = — TT- 



Therefore a + & = -g = -5. Hence a(a-\-h)=—„ and &(« + &) = -, . 



x^ y^ ^ ^ x' y 



a* &* 

 Whence by addition (a + 6)^ = — 3 + -j. 



Fig. 11. 



6^ 



Now we have by a general property of the ellipse CT = — and 00=—, whence 



X y 



a* 6* 

 0T^=— 2 +-2, which we have just proved to be equal to (« + 6f. Therefore we 



», ^ y 



have the curious result, 0T=« + J. 



Now, OT : OP : : CT : MP; 



or, a + l 



OP : : - 



X 



X. 



Therefore OP =(« + &) 



X 



And since at Fagnani's point -,'= 



it follows that 



a' ^ a" a + b' 



0P=:«. And similarly it is shown that FT=b. 



From whence the following curious theorem follows, — It is well known that 

 if OCT is a right angle, and OT is a line of given length, which moves so as to 

 keep its extremities constantly in the lines CO, CT, then any point P of the line 

 OT will generate an ellipse. But there is only one position of the line OT in 

 which it touches the ellipse. 



Theorem. — If the generating line OT touches the ellipse, the point of contact P 

 is Fagnani's point. 



Hence, if we let fall the perpendicular CX upon OT, then OX =PT= semi-axis 

 minor. For OF=a, and we have seen in the course of this investigation, that 



VX = a-b .-. 0X=0P-PX=6. 



The following corollary is also worth remarking. 



OP - arc BP = PT - arc PA. 

 VOL. XXIII. PART II. 4 L 



