MR TALBOT ON FAGNANI S THEOREM. 297 



In a similar way it is shown that yy^ = y^y^ , each side being the product of the 

 four semi-axes minor. 



Now, we wish to prove the equality of the expressions 



(.r, - xy + (2/1 -yf and {x^ - x^^ + (2/3 - t/j)^, 

 Subtract from them respectively the quantities 



- 2 {xx^ + yy^ and - 2 {x^x^ + 2/3^2) 

 which we have just proved to be equal. 

 The remainders will be sc^^-x.^-vy^+y'^, 



And -^2^ + ^3" + 2/2^ +2/3"- 



And now it is required to prove that these two remainders are equal. 

 But we have proved in theorem B, that 



«2+2/2 = ^2^^2_i . and by similar reasoning, 



Therefore (.2?^+^^) + (.2?/ + ?/i^) = — 2 + the sum of the squares of the 4 semi-axes 

 major. 



And by similar reasoning, {oc^ ^yi) -ir {00^ -^ yi^ is equal to the same quantity. 

 Therefore the two remainders are equal ; and therefore the theorem is demon- 

 strated. 



From this theorem several others may be deduced, by giving extreme values 

 to the four curves. 



In the first place, if the two ellipses are drawn infinitely near to each other, 

 and likewise the two hyperbolas infinitely near to each other, then, because con- 

 focal conies always intersect at right angles, the small quadrilateral formed by 

 the four intersections will be a rectangle, and of course its diagonals will be 

 equal. 



Next suppose that the axis minor of the first ellipse is infinitely diminished, 

 the quadrant of the curve will be reduced to the straight line CH, extending from 

 the centre to the focus. At the same time let the semi-axis major of the first 

 hyperbola be infinitely diminished, and the vertex of it will then coincide with 

 the centre, and the curve itself will become a straight line in the direction of the 

 axis minor produced to infinity. The four curves will thus be reduced to one 

 ellipse and one hyperbola, and two rectangular straight lines. The quadrilateral 

 figure then becomes BCVP (see figure 14), and our theorem asserts that in that 

 case CP = BV, the truth of which was independently proved in theorem B. 



Now, let the other hyperbola also have its axis minor infinitely diminished, its 

 vertex will then coincide with the focus, and the curve will be reduced to the 

 straight line HA produced to infinity. The line CP then becomes CA, and BV 



VOL. XXIII. PART II. 4 M 



