398. On an infinite line connecting two light sources, find the point equally illuminated by both sources.

## Simpson's Problem.
398. This problem, first published in Clairaut's "Algebra," has been passed on to textbooks under the name of the problem of light sources. Let the distance between the sources $A$ and $B$ be $d$. The distance of the sought point $C$ (equally illuminated) from the source $A$ is $x$. Then the distance to the other source $B = d - x$. If the first source has a strength of $a$ candles, and the second has a strength of $b$ candles, then

$$
\frac{a}{x^{2}}=\frac{b}{(d-x)^{2}}
$$

from which

$$
x^{2}(a-b)-2 a d x + a d^{2}=0
$$

and

$$
x = d \sqrt{a} \cdot \frac{\sqrt{a} \pm \sqrt{b}}{a-b}
$$

so that

$$
\begin{aligned}
& x_{1} = \frac{d \sqrt{a}}{\sqrt{a} - \sqrt{b}} \\
& x_{2} = \frac{d \sqrt{a}}{\sqrt{a} + \sqrt{b}}
\end{aligned}
$$

This problem is very convenient for investigation. Let's limit ourselves to indicating the course of the investigation:

$$
\begin{aligned}
& \text{for } d > 0 \text{ 1) } a > b \\
& \text{2) } a < b \\
& \text{3) } a = b \\
& \text{for } d = 0 \text{ 1) } a \neq b \\
& \text{2) } a = b
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-173.jpg?height=591&width=514&top_left_y=1555&top_left_x=380)

Fig. 88.

Simpson, Robert (1687-1768), English mathematician, professor at the University of Glasgow, who worked exclusively in the field of geometry.