If the radiant heat absorbed by the body is 75\% of the incident...

Let the total incident heat be $Q$.

The radiant heat absorbed by the body is $75\%$ of the incident heat. Therefore, the coefficient of absorption is $a = 0.75$.

The coefficient of reflection is given as $r = 0.15$.

According to the conservation of energy, the sum of the coefficients of absorption, reflection, and transmission is equal to $1$.

$a + r + t = 1$

Substituting the given values:

$0.75 + 0.15 + t = 1$

$0.90 + t = 1$

$t = 1 - 0.90 = 0.10$

The coefficient of transmission is $0.10$.

Answer: $0.10$