The main purpose of the present paper is to prove the theorem of RiemannRoch for adjoint systems on 3-dimensional algebraic varieties' by means of the theory of harmonic integrals. Let 9M be a non-singular algebraic variety of dimension 3 imbedded in a projective space, K be a canonical divisor on 911 and 8 be an arbitrary (possibly reducible) surface on 9b with only ordinary singularities i.e. a double curve i on which there is a finite number of ordinary cuspoidal points and of triple points of S. The problem concerning the theorem of Riemann-Roch for the adjoint system I K + 8 I of 8 consists in expressing the dimension of K + S I in terms of the virtual characters2 of K + S and other numerical characteristics of S, e.g. the number of linearly independent simple or double differentials of the first kind on SD vanishing on S, the number of connected components of S, etc. First, in Sections 1 to 7, we shall consider a more general case in which 9i is a 3-dimensional compact Kahlerian manifold and, using the theory of harmonic integrals,3 prove a formula expressing the number of linearly independent meromorphic triple differentials on 9W which are multiples of S in terms of the virtual arithmetic genus a(S) of S, the constant a(N) defined as a(J) = r3 r2 + ri , r, being the number of linearly independent v-ple differentials of the first, kind on a1, and other numerical characteristics of S. In the following Section 8, we shall return to the case in which 9) is algebraic and show that the dimension of IK + S I is equal to a(S) + a(g) 1 if the complete linear system S I partially contains a hyperplane section E of 9W in the sense that S E I] contains a surface with ordinary singularities only. In Section 9, we shall prove that the virtual arithmetic genus a(S) defined in terms of the arithmetic genera of irreducible components of S and of several numerical characteristics of the singularities of S is represented as a(S) = (2S3 + 3KS2 + K2S + CS)/12 1, where C is the canonical curve on 9M and S3, KS2, ... denote the topological intersection numbers I(S, S, S), I(K, S, S), -.. . Then we shall define the virtual arithmetic genus a(D) of an arbitrary divisor D on 9N by setting a(D) = (2D3 + 3KD2 + K2D + CD)/12 1. In Section 10, we shall prove the theorem of Riemann-Roch for the adjoint system I D I = I K + S I of S which says that4 dim I D = r(D2) + a(D) -Pa(9) + 3 mn + 1 k + 6 + '7, where r(D2) = D3 + 2 KD2 + 1, pa(9N) =KC/12 a() + 2, m is the num-