Abstract: Gaierkin method is used to solve the restricted integral equation arising from elastic torsion of multihole bars.We prove that the corresponding variational problem and discrete variational problem are uniquely solvable.The coefficient matrix of the linear equations is nonpositive definite because of the restrictions,but we prove the leading principal minors of the matrix do not equal zero and the triangular decomposition or Gaussian elimination can be used to the linear equations.A numerical example is given which shows the numerical results are satisfactory.