Find eight positive integers $n_1, n_2, \dots , n_8$ with the following property:

For every integer $k$, $-1985 \leq k \leq 1985$, there are eight integers $a_1, a_2, \dots, a_8$, each belonging to the set $\{-1, 0, 1\}$, such that $k=\sum_{i=1}^{8} a_i n_i .$
To solve this problem, we need to find eight positive integers \( n_1, n_2, \dots, n_8 \) such that for every integer \( k \) in the range \(-1985 \leq k \leq 1985\), there exist integers \( a_1, a_2, \dots, a_8 \) each belonging to the set \(\{-1, 0, 1\}\) such that \( k = \sum_{i=1}^{8} a_i n_i \).

1. **Choosing the integers**:
   We need to choose \( n_1, n_2, \dots, n_8 \) such that the sum of any combination of these integers, with coefficients from \(\{-1, 0, 1\}\), can cover the entire range from \(-1985\) to \(1985\).

2. **Using powers of 3**:
   Let's consider the integers \( n_i = 3^{i-1} \) for \( i = 1, 2, \dots, 8 \). This means:
   \[
   n_1 = 1, \quad n_2 = 3, \quad n_3 = 9, \quad n_4 = 27, \quad n_5 = 81, \quad n_6 = 243, \quad n_7 = 729, \quad n_8 = 2187
   \]

3. **Range of sums**:
   The maximum possible sum when all \( a_i = 1 \) is:
   \[
   \sum_{i=1}^{8} 3^{i-1} = 1 + 3 + 9 + 27 + 81 + 243 + 729 + 2187 = 3280
   \]
   Similarly, the minimum possible sum when all \( a_i = -1 \) is:
   \[
   \sum_{i=1}^{8} (-3^{i-1}) = -1 - 3 - 9 - 27 - 81 - 243 - 729 - 2187 = -3280
   \]

4. **Covering the range**:
   Since the range \([-3280, 3280]\) covers the range \([-1985, 1985]\), we can conclude that any integer \( k \) in the range \(-1985 \leq k \leq 1985\) can be expressed as:
   \[
   k = \sum_{i=1}^{8} a_i n_i \quad \text{where} \quad a_i \in \{-1, 0, 1\}
   \]

Thus, the integers \( n_1, n_2, \dots, n_8 \) that satisfy the given property are \( 1, 3, 9, 27, 81, 243, 729, 2187 \).

The final answer is \( \boxed{ 1, 3, 9, 27, 81, 243, 729, 2187 } \)