Recent applications in the data science and wireless communications give rise to a particular Rayleigh-quotient maximization, namely, maximizing the sum-of-Rayleigh-quotients over a sphere constraint. Previously, it is shown that maximizing the sum of two Rayleigh quotients is related with a certain eigenvector-dependent nonlinear eigenvalue problem (NEPv), and any global maximizer must be an eigenvector associated with the largest eigenvalue of this NEPv. Based on such a principle for the global maximizer, the self-consistent field (SCF) iteration turns out to be an efficient numerical method. However, generalization of sum of two Rayleigh-quotients to the sum of an arbitrary number of Rayleigh-quotients maximization is not a trivial task. In this paper, we shall develop a new treatment based on the S-Lemma. The new argument, on one hand, handles the sum of two and three Rayleigh-quotients maximizations in a simple way, and also deals with certain general cases, on the other hand. Our result gives a characterization for the solution of this sum-of-Rayleigh-quotients maximization and provides theoretical foundation for an associated SCF iteration. Preliminary numerical results are reported to demonstrate the performance of the SCF iteration.