Abstract: Correlations between the elliptic or triangular flow coefficients v(m) (m = 2 or 3) and other flow harmonics v(n) (n = 2 to 5) are measured using root S-NN = 2.76 TeV Pb + Pb collision data collected in 2010 by the ATLAS experiment at the LHC, corresponding to an integrated luminosity of 7 μb(-1). The v(m)-v(n) correlations aremeasured in midrapidity as a function of centrality, and, for events within the same centrality interval, as a function of event ellipticity or triangularity defined in a forward rapidity region. For events within the same centrality interval, v(3) is found to be anticorrelated with v(2) and this anticorrelation is consistent with similar anticorrelations between the corresponding eccentricities, epsilon(2) and epsilon(3). However, it is observed that v(4) increases strongly with v(2), and v(5) increases strongly with both v(2) and v(3). The trend and strength of the v(m) -v(n) correlations for n = 4 and 5 are found to disagree with epsilon(m)-epsilon(n) correlations predicted by initial-geometry models. Instead, these correlations are found to be consistent with the combined effects of a linear contribution to v(n) and a nonlinear term that is a function of v(2)(2) or of v(2)v(3), as predicted by hydrodynamic models. A simple two-component fit is used to separate these two contributions. The extracted linear and nonlinear contributions to v(4) and v(5) are found to be consistent with previously measured event-plane correlations.