Searching for a mathematical justification of ``Gompertz's law of mortality'' and of other `noteworthy fundamental formulas' of actuarial mathematics, \textit{A. De Morgan} [Assurance Mag. J. Inst. Actuaries 81, 181-184 (1859)] and \textit{M. Chini} [Periodico di Mat. (3) 4, 264-270 (1907; JFM 38.0373.02)] determined the differentiable solutions to the functional equation \(f(x+y)+f(x+z)=cf[x+h(y,z)].\) The present authors consider that equation over real intervals of positive lengths (\(I\) for \(x,\) \(J\) for \(y\) and \(z\), rather than over \(\mathbb{R}^3\) for \((x,y,z)\)) and, by application of two results of \textit{C. T. Ng} [Proc. Am. Math. Soc. 39, 525-529 (1973; Zbl 0272.39009) and Ann. Polon. Math. 27, 329-336 (1973; Zbl 0251.39003)] offer all real valued solutions for which \(f\) is locally bounded and \(h\) is continuous in each variable. By specialization of a result of \textit{A. Lundberg} [Aequationes Math. 16, 21-30 (1977; Zbl 0433.39011)] the general continuous solutions of the more general equation \(f(x)+f(x+z)=cf(x+g(z))\) can be obtained on \(\mathbb{R}^2\) if \(f\) is also supposed to be ``philandering'', i.e. nonconstant on every subinterval. In the present paper a lengthier direct proof is offered without the ``philandering'' assumption. [Remark: The one-time appearence of \(\varphi\) in the ninth line of section 2 (presumably it should be \(f_x\)) and the three-digit labelling of Lemmas, Propositions, Theorems, Corollaries, and Remarks (the second digit is the same as the first and is omitted in references), that could be the journal's fault rather than the authors', may be confusing to the reader].