The transverse vibration of a type of slender beam of continuously varying rectangular cross-section is considered in this paper. One side of the cross-section is constant, while the other side is proportional to the square root of the axial co-ordinate. It is assumed that the beam dimensions are such that application of the Euler-Bernoulli theory of bending is valid. Thus the vibration in the two principal planes will be uncoupled. The two mode shape equations are fourth order linear differential equations with variable coefficients and with regular singularity. Analytical solutions of the two mode shape equations are derived, based on the method of Frobenius. Combinations of ideal clamped, pinned, sliding and free boundary conditions are considered. The sharp end of a "complete" beam cannot sustain a bending moment or shearing force and hence must always remain free. The large end may be clamped, pinned, sliding or free. The frequency equations for a "complete" beam is the determinant of a 2 × 2 matrix equated to zero. For a truncated beam any combinations of the four boundary conditions is possible and the frequency equations are the determinant of a 4 × 4 matrix equated to zero. The first three dimensionless natural frequencies for vibration in the two principal planes are tabulated for 16 combinations of the four boundary conditions and truncation factors from 0·05 to 0·7. To preserve the accuracy of the frequencies to six figures, the results are presented in tabular form. It is hoped that they will serve as benchmarks.