We consider a Finsler space $F^{n}$ equipped with a fundamental function $L(x, y)$. Let $g(x, y)$ be the determinant consisting of components $g_{ij}(x, y)$ of the fundamental tensor of $F^{n}$. We sometimes have experience giving us to understand some importance of the scalar ${}^{*}L =Lg^{w/2}$ as it will be reported in §2. Thus it seems to the present author that a theory of Finsler spaces based on this scalar ${}^{*}L(x, y)$ may come in useful. The main purpose of the present paper is to construct metrical Finsler connections from ${}^{*}L(x, y)$.