Let $\eta = Q(\xi )$ be the polynomial transformation of the random variable $\xi $. The following rule is introduced in this article in order to calculate the semi-invariants of $\eta $ from the semi-invariants of $\xi $.It is necessary 1. to express the moments of $\eta $ in terms of those of $\xi $ according to (III), 2. to replace in (III) the expression for the moments of $\xi $with their semi-invariants according to (I.a), 3. to cancel some terms in the expression obtained according to the law formulated in the theorem.By employing this rule in § 4, we have calculated all the semi-invariants of the random function depending quadratically on Laplace’s function.