Several pairs of positive integers $(m ,n )$ satisfy the condition $19m + 90 + 8n = 1998$.  Of these, $(100, 1 )$  is the pair with the smallest value for $n$. Find the pair with the smallest value for $m$.
1. Start with the given equation:
   \[
   19m + 90 + 8n = 1998
   \]
   Subtract 90 from both sides to simplify:
   \[
   19m + 8n = 1908
   \]

2. Rearrange the equation to solve for \(19m\):
   \[
   19m = 1908 - 8n
   \]
   Since \(m\) and \(n\) are positive integers, \(1908 - 8n\) must be divisible by 19. Therefore, we need:
   \[
   1908 - 8n \equiv 0 \pmod{19}
   \]
   Simplify the congruence:
   \[
   1908 \equiv 8n \pmod{19}
   \]

3. Calculate \(1908 \mod 19\):
   \[
   1908 \div 19 = 100 \quad \text{with a remainder of} \quad 1908 - 19 \times 100 = 8
   \]
   Thus:
   \[
   1908 \equiv 8 \pmod{19}
   \]
   So the congruence becomes:
   \[
   8n \equiv 8 \pmod{19}
   \]

4. Solve for \(n\):
   \[
   8n \equiv 8 \pmod{19}
   \]
   Divide both sides by 8 (since 8 and 19 are coprime, 8 has a multiplicative inverse modulo 19):
   \[
   n \equiv 1 \pmod{19}
   \]
   Therefore, \(n\) can be written as:
   \[
   n = 19k + 1 \quad \text{for some integer } k
   \]

5. Substitute \(n = 19k + 1\) back into the equation \(19m + 8n = 1908\):
   \[
   19m + 8(19k + 1) = 1908
   \]
   Simplify:
   \[
   19m + 152k + 8 = 1908
   \]
   \[
   19m + 152k = 1900
   \]

6. Solve for \(m\):
   \[
   19m = 1900 - 152k
   \]
   \[
   m = \frac{1900 - 152k}{19}
   \]
   Simplify the fraction:
   \[
   m = 100 - 8k
   \]

7. Ensure \(m\) and \(n\) are positive integers:
   \[
   100 - 8k > 0 \quad \Rightarrow \quad k < 12.5
   \]
   Since \(k\) must be an integer, the largest possible value for \(k\) is 12. To minimize \(m\), we choose the largest \(k\):
   \[
   k = 12
   \]
   Substitute \(k = 12\) back into the expressions for \(m\) and \(n\):
   \[
   m = 100 - 8 \times 12 = 100 - 96 = 4
   \]
   \[
   n = 19 \times 12 + 1 = 228 + 1 = 229
   \]

Conclusion:
The pair \((m, n)\) with the smallest value for \(m\) is \((4, 229)\).

The final answer is \( \boxed{ (4, 229) } \).