Part I Introduction to the Study of the Elliptic Modular Functions 1 On the invariants of the binary biquadratic form 1．1 The form f(z1， z2) and its irrational invariants 1．2 The irrational invariants A， B， C of the form f 1．3 Behavior of the A， B， C upon varying the sequence of factors of f 1．4 Equivalence of two forms with the same sequence of factors． First canonical form of f 1．5 The equivalence of a form f with itself 1．6 The equivalence of two forms with arbitrary factor sequence 1．7 The rational invariants of the form f 1．8 The rational invariants in explicit form 1．9 The invariants g2， g3 and the absolute invariant J 1．10 The second canonical form off 1．11 Geometrical observations on the second canonical form 1．12 Third conical form off 1．13 More on the third canonical form． Connection to the theory of the regular solids 1．14 Normal forms of the elliptic integral of the first kind 1．15 Naming the normal forms． History 2 On the periods of the elliptic integral of the first kind 2．1 Pairs of primitive periods of the integral of the first kind 2．2 The periods as invariants． Dependency on the rational invariants．Normalization of the periods 2．3 Setting up the differential equation for the normalized periods 2．4 Fundamental theorems concerning the dependency of the normalized periods on J 2．5 Choice of a special primitive period-pair 2．6 Dissection of the J-plane． Significance of the determinations of the previous paragraph 2．7 Approach to the neighborhood of a singular point 2．8 Preliminary determination of the numbers kl， k2 2．9 Carrying though the investigation for the singular point J = 0 2．10 Carrying the investigation through for the singular point J = 1 2．11 Determinations for the neighborhood of J = oo and associated calculation of Ω2 2．12 Calculation of the limiting value offal for J —— oo 2．13 Disposal of the singular point J = oo． Historical remarks 2．14 Branching of the periods tobto2 over the/-plane 2．15 The periodquotient to as a function ofJ 2．16 Differential equation of the third order for to(J)． The s-functions Concerning certain conformal mappings and the triangle functions arising from them 3．1 Replacement of the Riemann surface occurring by simpler figures 3．2 Figure for the representation of the connection between A and J 3．3 Carrying the λ-plane onto the surface of the sphere 3．4 Relation to the second chapter 3．5 Figures for illustrating the connection between u and J 3．6 The mapping of a circular arc triangle onto the halfplane of J 3．7 The relation of A to u illustrated through figures 3．8 The circle-relation． Theorems on circular arc triangles 3．9 The symmetry with respect to a circle 3．10 The law of symmetry． Direct and indirect circle-relatedness 3．11 Significance of the law of symmetry for the function u(J) 3．12 General investigation of the function-theoretic significance of the law of symmetry 3．13 Definition and fundamental properties of the triangle- or s-functions 3．14 Series developments for a branch of the s-function 3．15 Differential equation of the third order for the s-function 3．16 Assembly of the triangle functions already appearing 3．17 Division of the triangle functions into kinds 3．18 The s-functions of the first kind 3．19 The s-functions of the second kind 3．20 The s-functions of the third kind 3．21 The triangle figures associated to w(λ) and w(J) 4 Development of the definitions and fundamental problems of a theory of the elliptic modular functions 4．1 The Legendre relation 4．2 The rational invariants g2， g3，△ as functions of the periods to1， to2 4．3 Functional determinants of the forms g2， g3， △ 4．4 The periods of integrals of the second kind as functions of w1， w2 4．5 Calculation of the Hessian determinant H(logA) 4．6 Mappings effected by to(J) and ((J)． Icosahedral and modular equation Part II Treatment of the Group-Theoretic Fundamental Problem Part III The Function-Theoretic Fundamental Problem Commentaries
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