On a Method of Calculation of Semi-Invariants

Let = Q () be the polynomial transformation of the random variable. The following rule is introduced in this article in order to calculate the semi-invariants of from the semi-invariants of. It is necessary 1. to express the moments of in terms of those of according to (III), 2. to replace in (III) the expression for the moments of 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.