 Determine all sequences of equal ratios of the form 

     \[
     \frac{a_1}{a_2} = \frac{a_3}{a_4} = \frac{a_5}{a_6} = \frac{a_7}{a_8}
     \]

     which  simultaneously satisfy the following conditions:

     $\bullet$ The set $\{ a_1, a_2, \ldots , a_8 \}$ represents all positive divisors of $24$.

     $\bullet$  The common value of the ratios is a natural number.
To determine all sequences of equal ratios of the form 

\[
\frac{a_1}{a_2} = \frac{a_3}{a_4} = \frac{a_5}{a_6} = \frac{a_7}{a_8}
\]

which simultaneously satisfy the given conditions, we need to follow these steps:

1. **Identify the positive divisors of 24:**
   The positive divisors of 24 are \(1, 2, 3, 4, 6, 8, 12, 24\).

2. **Set up the ratios:**
   We need to find sequences of the form \(\frac{a_1}{a_2} = \frac{a_3}{a_4} = \frac{a_5}{a_6} = \frac{a_7}{a_8}\) where each ratio is a natural number and all \(a_i\) are distinct positive divisors of 24.

3. **Check possible ratios:**
   Since the common value of the ratios is a natural number, let’s denote this common ratio by \(k\). Therefore, we have:
   \[
   \frac{a_1}{a_2} = \frac{a_3}{a_4} = \frac{a_5}{a_6} = \frac{a_7}{a_8} = k
   \]
   This implies:
   \[
   a_1 = k \cdot a_2, \quad a_3 = k \cdot a_4, \quad a_5 = k \cdot a_6, \quad a_7 = k \cdot a_8
   \]

4. **List possible values of \(k\):**
   Since \(k\) is a natural number, we need to check which values of \(k\) allow \(a_i\) to be distinct positive divisors of 24. The possible values of \(k\) are those that can be formed by dividing one divisor of 24 by another.

5. **Check each possible \(k\):**
   - For \(k = 1\):
     \[
     \frac{1}{1}, \frac{2}{2}, \frac{3}{3}, \frac{4}{4}, \frac{6}{6}, \frac{8}{8}, \frac{12}{12}, \frac{24}{24}
     \]
     This does not work since all \(a_i\) must be distinct.

   - For \(k = 2\):
     \[
     \frac{2}{1}, \frac{4}{2}, \frac{6}{3}, \frac{8}{4}, \frac{12}{6}, \frac{24}{12}
     \]
     This works and gives the sequence:
     \[
     \frac{2}{1} = \frac{4}{2} = \frac{6}{3} = \frac{8}{4} = 2
     \]

   - For \(k = 3\):
     \[
     \frac{3}{1}, \frac{6}{2}, \frac{12}{4}, \frac{24}{8}
     \]
     This works and gives the sequence:
     \[
     \frac{3}{1} = \frac{6}{2} = \frac{12}{4} = \frac{24}{8} = 3
     \]

   - For \(k = 4\):
     \[
     \frac{4}{1}, \frac{8}{2}, \frac{24}{6}
     \]
     This does not work since we need four pairs.

   - For \(k = 6\):
     \[
     \frac{6}{1}, \frac{12}{2}, \frac{24}{4}
     \]
     This does not work since we need four pairs.

   - For \(k = 8\):
     \[
     \frac{8}{1}, \frac{24}{3}
     \]
     This does not work since we need four pairs.

   - For \(k = 12\):
     \[
     \frac{12}{1}, \frac{24}{2}
     \]
     This does not work since we need four pairs.

   - For \(k = 24\):
     \[
     \frac{24}{1}
     \]
     This does not work since we need four pairs.

6. **Conclusion:**
   The only possible sequences that satisfy the conditions are for \(k = 2\) and \(k = 3\).

The final answer is \(\boxed{\left\{ \frac{2}{1} = \frac{4}{2} = \frac{6}{3} = \frac{8}{4} = 2 \right\}, \left\{ \frac{3}{1} = \frac{6}{2} = \frac{12}{4} = \frac{24}{8} = 3 \right\}}\)