The eigenvalue problem of the form \((A+\varepsilon B)\phi(\varepsilon)= \lambda(\varepsilon) \phi(\varepsilon)\) is considered, where \(A, B\) are matrices, or more generally, differential operators. If \(A\) is selfadjoint and the eigenvalue \(\lambda(0)\) of \(A\) is simple, it is shown that \(\lambda(\varepsilon)\) has a power series expansion \(\lambda(\varepsilon)=\lambda(0)+\varepsilon \mu_1+\varepsilon^2\mu_2+\cdots\), and formulas to express the coefficients \(\mu_1\) and \(\mu_2\) are obtained. As an application, the vibration of a membrane under a small perturbation \(\varepsilon\), which results in a shape of a trapezoid is studied.