Abstract. Mathieu's differential equation, y" + (a -2q cos 2x)y = 0, admits of solutions of period ir or 2x for four countable sets of characteristic values, aiq), which can be ordered as ariq), r = 0, 1, • • -. The power series expansions for the ariq) converge up to the first double point for that order in the complex plane. [At a double point, ar(g) = ar+2iq).] The present work furnishes the double points for orders r up to and including 15. These double points are singular points, and the usual methods of determining the characteristic values break down at a singular point. However, it was possible to determine two smooth functions in which one could interpolate for both q and aTiq) at the singular point. The method is quite general and can be used in other problems as well. |

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