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 √ 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 are measured 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, 2 and 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 m -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.