The membrane has a shape similar to a catenary of uniform strength. When φ is zero or negligible, the equations reduce to the usual Poisson equations for thin plates. φ can be zero if the condition ( ∂²w/∂x∂y )² - (∂²w/∂x²)(∂²w/∂y²) = 0 is satisfied, which means the middle surface is a developable surface. When φ is zero, the middle surface is unstretched, and a plane sheet can be bent into a developable surface without stretching or shrinking. If the left-hand side of equation (106) is small but not zero, φ can still be small, making Poisson's equations approximately true. This condition is satisfied if w is small compared to the plate's thickness. φ depends on w² rather than w. The condition that φ is negligible is satisfied if the deflection of the middle surface from a developable surface is small compared to the plate's thickness. An example of this is given in the third problem. In a paper on transverse vibrations of bars, the equation for transverse vibrations of a prismatic bar is corrected to account for "rotatory inertia" and shear deflection. The corrected equation is EI∂⁴y/∂x⁴ + (ρΩ/g)∂²y/∂t² = 0. The results for a rectangular bar are compared with previous work, confirming earlier conclusions. For plane strain, the equations are solved, and the solutions are expressed in terms of hyperbolic functions. The frequency equation is derived, and the velocity of waves is calculated. For long waves, the velocity approaches a limit, which is the velocity of Rayleigh waves. For short waves, the velocity is calculated using an approximate solution. The results are compared with exact solutions for circular cross-sections, showing good agreement. The paper concludes that the approximate solution can be applied to other cross-sections.The membrane has a shape similar to a catenary of uniform strength. When φ is zero or negligible, the equations reduce to the usual Poisson equations for thin plates. φ can be zero if the condition ( ∂²w/∂x∂y )² - (∂²w/∂x²)(∂²w/∂y²) = 0 is satisfied, which means the middle surface is a developable surface. When φ is zero, the middle surface is unstretched, and a plane sheet can be bent into a developable surface without stretching or shrinking. If the left-hand side of equation (106) is small but not zero, φ can still be small, making Poisson's equations approximately true. This condition is satisfied if w is small compared to the plate's thickness. φ depends on w² rather than w. The condition that φ is negligible is satisfied if the deflection of the middle surface from a developable surface is small compared to the plate's thickness. An example of this is given in the third problem. In a paper on transverse vibrations of bars, the equation for transverse vibrations of a prismatic bar is corrected to account for "rotatory inertia" and shear deflection. The corrected equation is EI∂⁴y/∂x⁴ + (ρΩ/g)∂²y/∂t² = 0. The results for a rectangular bar are compared with previous work, confirming earlier conclusions. For plane strain, the equations are solved, and the solutions are expressed in terms of hyperbolic functions. The frequency equation is derived, and the velocity of waves is calculated. For long waves, the velocity approaches a limit, which is the velocity of Rayleigh waves. For short waves, the velocity is calculated using an approximate solution. The results are compared with exact solutions for circular cross-sections, showing good agreement. The paper concludes that the approximate solution can be applied to other cross-sections.