A rectangular pool table has a hole at each of three of its corners. The lengths of sides of the table are the real numbers $a$ and $b$. A billiard ball is shot from the fourth corner along its angle bisector. The ball falls in one of the holes. What should the relation between $a$ and $b$ be for this to happen?
To solve this problem, we need to analyze the trajectory of the billiard ball shot from the fourth corner along its angle bisector and determine the relationship between \(a\) and \(b\) such that the ball falls into one of the holes.

1. **Setup the Problem:**
   - Consider a rectangular pool table with sides of lengths \(a\) and \(b\).
   - The table has holes at three of its corners, say at \((0,0)\), \((a,0)\), and \((0,b)\).
   - The ball is shot from the fourth corner \((a,b)\) along the angle bisector of the right angle at \((a,b)\).

2. **Angle Bisector Trajectory:**
   - The angle bisector of the right angle at \((a,b)\) will have a slope of \(-1\) because it equally divides the 90-degree angle into two 45-degree angles.
   - Therefore, the equation of the line along which the ball is shot can be written as:
     \[
     y - b = -1(x - a) \implies y = -x + a + b
     \]

3. **Reflection and Path Analysis:**
   - To understand the path of the ball, we can use the method of reflecting the table across its sides.
   - Reflect the table repeatedly with respect to the vertical lines \(x = a, 2a, 3a, \ldots\) and horizontal lines \(y = b, 2b, 3b, \ldots\).
   - This creates an infinite grid of reflected tables, and the ball's path will be a straight line in this grid.

4. **Intersection with Holes:**
   - The ball will fall into a hole if it intersects one of the lines \(x = ka\) or \(y = lb\) for some integers \(k\) and \(l\).
   - The line \(y = -x + a + b\) intersects the vertical lines \(x = ka\) at points \((ka, -ka + a + b)\).
   - For the ball to fall into a hole, \(-ka + a + b\) must be a multiple of \(b\), i.e., \(-ka + a + b = lb\) for some integer \(l\).

5. **Solving for \(a\) and \(b\):**
   - Rearrange the equation \(-ka + a + b = lb\):
     \[
     a(1 - k) + b = lb
     \]
     \[
     a(1 - k) = lb - b
     \]
     \[
     a(1 - k) = b(l - 1)
     \]
   - For this equation to hold for some integers \(k\) and \(l\), \(a\) and \(b\) must be such that the ratio \(\frac{a}{b}\) is rational. Let \(\frac{a}{b} = \frac{m}{n}\) for some integers \(m\) and \(n\).

6. **Conclusion:**
   - The relationship between \(a\) and \(b\) must be such that \(\frac{a}{b}\) is a rational number. This ensures that the ball will eventually fall into one of the holes.

\(\blacksquare\)

The final answer is \( \boxed{ \frac{a}{b} } \) must be rational.