To solve the problem of finding the integers \( x, y, z \) such that \( P^x Q^y C^z \) is dimensionless, we start by identifying the dimensions of each variable involved:
1. **Identify the dimensions:**
- Radiation pressure \( P \): The dimension of pressure is \( [M L^{-1} T^{-2}] \).
- Radiation energy per unit area per second \( Q \): This is equivalent to power per unit area, which has the dimension \( [M L^{-3} T^{-3}] \).
- Speed of light \( C \): The dimension of speed is \( [L T^{-1}] \).
2. **Express the dimensions in terms of \( M, L, T \):**
- For \( P \):
\[
[P] = [M L^{-1} T^{-2}]
\]
- For \( Q \):
\[
[Q] = [M L^{-3} T^{-3}]
\]
- For \( C \):
\[
[C] = [L T^{-1}]
\]
3. **Set up the equation for dimensional analysis:**
We need to find \( x, y, z \) such that:
\[
P^x Q^y C^z \text{ is dimensionless}
\]
This means:
\[
[P^x Q^y C^z] = [M^0 L^0 T^0]
\]
4. **Substituting the dimensions:**
Substituting the dimensions into the equation gives:
\[
[M^{x} L^{-x} T^{-2x}] \cdot [M^{y} L^{-3y} T^{-3y}] \cdot [L^{z} T^{-z}]
\]
Combining these, we have:
\[
[M^{x+y} L^{-x-3y+z} T^{-2x-3y-z}]
\]
5. **Setting up the equations:**
For the expression to be dimensionless, the exponents of \( M, L, T \) must all equal zero:
- From \( M \):
\[
x + y = 0 \quad \text{(1)}
\]
- From \( L \):
\[
-x - 3y + z = 0 \quad \text{(2)}
\]
- From \( T \):
\[
-2x - 3y - z = 0 \quad \text{(3)}
\]
6. **Solving the equations:**
- From equation (1), we can express \( y \) in terms of \( x \):
\[
y = -x
\]
- Substitute \( y = -x \) into equation (2):
\[
-x - 3(-x) + z = 0 \implies -x + 3x + z = 0 \implies 2x + z = 0 \implies z = -2x \quad \text{(4)}
\]
- Substitute \( y = -x \) and \( z = -2x \) into equation (3):
\[
-2x - 3(-x) - (-2x) = 0 \implies -2x + 3x + 2x = 0 \implies 3x = 0
\]
This does not provide new information, so we use the relationships from (1) and (4).
7. **Choosing a value for \( x \):**
Let’s choose \( x = 1 \):
- Then from (1): \( y = -1 \)
- From (4): \( z = -2 \)
Thus, we find:
\[
x = 1, \quad y = -1, \quad z = -2
\]
8. **Final answer:**
The values of \( x, y, z \) are:
\[
x = 1, \quad y = -1, \quad z = -2
\]
