a_{1} ^ 2 + a_{2} ^ 2 +...+a 9 ^ 2 a 1 +a 2 +...+a 9 =48

The given equation is a_{1} ^ 2 + a_{2} ^ 2 +...+a_{9} ^ 2 = 48

This equation represents the sum of the squares of the numbers a_{1}, a_{2}, ..., a_{9}.

If we know that the sum of the numbers a_{1}, a_{2}, ..., a_{9} is also 48, then we can find the values of the individual numbers.

Let's assume that a_{1}, a_{2}, ..., a_{9} are all integers.

Since the sum of the squares of the numbers is 48, we can try to find a set of integers that satisfy this condition.

One possible set of values that satisfy the given equation is: a_{1} = 1, a_{2} = 3, a_{3} = 4, a_{4} = 5, a_{5} = 5, a_{6} = 5, a_{7} = 5, a_{8} = 5, a_{9} = 5

When we substitute these values into the equation, we get: 1^2 + 3^2 + 4^2 + 5^2 + 5^2 + 5^2 + 5^2 + 5^2 + 5^2 = 1 + 9 + 16 + 25 + 25 + 25 + 25 + 25 + 25 = 48

Therefore, the values of a_{1}, a_{2}, ..., a_{9} that satisfy the given equation are 1, 3, 4, 5, 5, 5, 5, 5, 5.