Let's inscribe a rectangle with the largest area in the ellipse $b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}(a>b)$ (the sides of the rectangle are parallel to the major axes of the ellipse). In this rectangle, inscribe an ellipse whose major axes are equal to the sides of the rectangle. In the thus obtained ellipse, inscribe a rectangle with the largest area, and so on. What is the sum of the areas of the infinitely many rectangles and ellipses thus obtained?
The ellipse is formed from the principal circle by an orthogonal affine transformation; the characteristic of the transformation is: $\frac{b_{⿹ 勹}}{a}$. The area of the principal circle: $a^{2} \pi$, the area of the ellipse: $a^{2} \pi \cdot \frac{b}{a}=a b \pi$.

Inscribe a square in the principal circle, whose sides are parallel to the major axes of the ellipse. The area of the square: $2 a^{2}$. The affine transformation of this square is the largest area rectangle inscribed in the ellipse (whose sides are parallel to the major axes). In the affine transformation, the side of the square inscribed in the circle that is parallel to the $X$-axis remains the same: $a \sqrt{2}$; the other side, however, is $\frac{b}{a} \cdot a \sqrt{2}=b \sqrt{2}$. The area of the rectangle is $a \sqrt{2} \cdot b \sqrt{2}=2 a b\left(=2 a^{2} \cdot \frac{b}{a}\right)$.

The major axis of the ellipse inscribed in this rectangle is: $a \sqrt{2}$, the minor axis is $b \sqrt{2}$. The area of the ellipse: $\frac{1}{2} a \sqrt{2} \cdot \frac{1}{2} b \sqrt{2} \cdot \pi=\frac{2 a b}{4} \pi=\frac{a b}{2} \pi$. The largest area rectangle inscribed in this ellipse is: $2 \cdot \frac{a \sqrt{2}}{2} \cdot \frac{b \sqrt{2}}{2}=a b$.

From this, we can already see that the areas of the ellipses in question form an infinite geometric series whose first term is $a b \pi$, and whose common ratio is $\frac{1}{2}$. The sum of these is therefore $2 a b \pi$.

The largest area rectangles also form such a series; the first term of the series is: $2 a b$, the common ratio is $\frac{1}{2}$. The sum of these is $4 a b$.[^0]


[^0]:    ${ }^{1}$ Vol. L. XL, p. 141. (Kárteszi: The Ellipse.) (May 1935, i.e., 1935/5 p. 141 - ed.)

    ${ }^{2}$ Vol. 8, No. 4, pp. 225-226. (1938/4 - ed.)