Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 3 10.wxyz/CV72P3 http://www.austms.org.au/Publ/Bulletin/V72P3/ Mark Harmer 13 February 2006 2006 2 13 385 389 723-5186-Harmer-2005 10.wxyz/C2005V72P3p385 http://www.austms.org.au/Publ/Bulletin/V72P3/723-5186-Harmer/ We show the rationality of the Taylor coefficients of the inverse of the Schwarz triangle functions for a triangle group about any vertex of the fundamental domain. 11F30, 42A16 Supported by a New Zealand FRST research fellowship. A.F. Beardon Springer-Verlag MR698777 A.F. Beardon; \textit{The geometry of discrete groups} (Springer-Verlag, Berlin, 1983). M. Harmer, G. Martin and B. Pavlov (Department of Maths Report series 499, University of Auckland, New Zealand, 2003) M. Harmer, G. Martin and B. Pavlov; Conformal mappings from the upper half plane to fundamental domains on the hyperbolic plane, (Department of Maths Report series 499, University of Auckland, New Zealand, 2003). J. Lehner Pacific J. Math. MR61168 J. Lehner; Note on the Schwarz triangle functions, \textit{Pacific J. Math.} \textbf{4} (1954), pp.~243--249. Z. Nehari McGraw-Hill MR45823 Z. Nehari; \textit{Conformal mapping} (McGraw-Hill, New York, 1952).