The first difference refers to the difference between consecutive terms in a sequence. Given the second term \( T_2 = 3a + 5 \) and the third term \( T_3 = 8a - 3 \), we can find the first difference \( D_1 \) as follows:
\[ D_1 = T_3 - T_2 \]
Substituting the values of \( T_2 \) and \( T_3 \):
\[ D_1 = (8a - 3) - (3a + 5) \]
Now, simplify the expression:
\[ D_1 = 8a - 3 - 3a - 5 \] \[ D_1 = (8a - 3a) + (-3 - 5) \] \[ D_1 = 5a - 8 \]
Thus, the first difference is:
\[ \boxed{5a - 8} \]