An effective method is proposed for solving the boundary integral equation (BIE) for the problem of a crack along a curvilinear surface in an elastic space on the basis of the transformation of the initial integrodifferential equation into an equation without derivatives. This is achieved by using the concept of the finite-part integral (FPI). Quadrature formulas are presented for such integrals over arbitrary convex polygons by approximating displacement discontinuities on the boundary by polynomials. The well-known BIE for three-dimensional cracks contain either derivatives of the unknown functions or derivatives of a surface integral /1-7/. In both cases the presence of the derivatives significantly complicates the solution. However, as is shown in /8/, these difficulties are reduced in the case of a plane crack of normal discontinuity if the FPI concept is utilized /9, 10/. In this connection, it is useful to investigate the possibility of applying such an approach to the more general problem of a crack of arbitrary discontinuity and to develop the numerical side of its utilization. Both aims are pursued in this paper: the extension of this idea to the general case of three-dimensional cracks is given and methods are indicated for evaluating the integrals that originate by presenting quadrature formulas convenient for the numerical realization of the BIE method on a computer.