WhenDarbouxprovedtheorthogonalityofsystemsofhomofocalovals,healsoshowedthatovalsprovideageometricalinterpretationoftheadditiontheoremandthattheyconstitutethealgebraicformoftheintegralsolution.
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When Darboux proved the orthogonality of systems of homofocal ovals, he also showed that ovals provide a geometrical interpretation of the addition theorem and that they constitute the algebraic form of the integral solution.