In consequence of the not inconsiderable circumstances， which have attended my working out of the modular functions， there presents itself a bit of a danger， that during the studying of the lectures and during the investigation of its principal concepts， that time was pressing or strength flagged. Certainly， the comprehensiveness of the present book is， in large part， a consequence of that breadth of the presentation， with which I have long occupied myself to clarify the attendant studies. On the other hand， other developments are presented as a kind of concession to other viewpoints of our subject， other investigations which have only the character of mere details， without the systematic study of which will not impair our general understanding. It is thereby not always easy for the beginner to immediately provide a motivation for the present investigation， and to place it into the total structure of this book; therefore I believe the usefulness of the latter is to allow me to indicate this with a wave of the hand.

　　In any case， I think it is useful to begin the study of the book with the first three chapters of the second section， whereby one will， if one wants， glance back at section 1 paragraphs 6 through 8 in chapter 2， having initially skipped them. At any rate， paragraph 9 will be understandable if one only knows the concept of modular substitution from part I. Soon the systematic studies of chapter I，3 would recommend themselves， insofar as the previous four chapters are seen as the real foundation of the theory of modular functions. If， moreover， there are， in the chapter I，3 just referred to， many relations partly to previous developments， and partly to the lectures on the icosahedron， I thus hope I have made these more inspiring， if perhaps harder to understand.

　　The reference to the theory of elliptic functions in its modern form leads to the essential circle of ideas of this book. In this sense， I cannot stress strongly enough， that paragraphs in I，l and 2， as well as part of 1， 4 should in no way be counted as fundamental， but only of passing interest. I can therefore only advise that one should， aftercompleting the four chapters II，1，2，3 and I，3， to familiarize oneself with the fundamental modular problems as they are developed on page 139ff.， in order to continue the

　　To find the relevant place， one avails oneself of the index at the end of the present volume. study in a similar manner. Moreover， one who knows the theory of the icosahedron will already obviously have acquired a foundation for understanding the fundamental problems in paragraphs 6 through 13 of chapter I，4. I certainly hope that by looking at chapter I，2 as well as at chapter I，4， one will have reviewed sufficiently to understand the theory of differential equations.

　　As absolutely essential investigations， I must now yet indicate those， which make up the contents of the two chapters II，4，5. Especially， the geometrical arguments and measures should be practiced until familiar; and it is a thoroughly mistaken conception， for me to accord geometry in the chapter in question only a secondary role， say only as a means of illustrating other analytic methods. The training required here in the geometrical treatment of group-theoretic and function-theoretic matters is also of the utmost importance for the theory of automorphic functions （for which our book is to prepare）.

　　Much earlierit was permissible to shorten the lectures in the two following chapters II，6，7; I can here designate as indispensable the first three paragraphs ofll，6 and the first five ofll，7. All others， as well as the two chapters II，8，9， have the character of particular calculations. Herewith， these special developments should have the significance， that there is no agreement that they are good in themselves; it has indeed become inadvisable， from now on， to pile technique upon technique， if we would not， at the same time， directly show their usefulness to our investigation. At the same time， one must， to some extent， leave to the reader， to make a choice among the group-theoretic details of the chapter in question， according to his own judgement and taste. It should yet be emphasized， that the semi-metacyclic subgroups on page 460 repeatedly play a role in the later parts of this work， and the wonderful theorem of Galois， formulated on page 489， together with its many applications （pages 646 and 749 in the first volume and page 419 in the second volume） are certainly the most dazzling results which the theory of modular functions has furnished.

　　Concerning the function-theoretic presentations， look at both chapters III，1，2 and then also their continuation in the second volume （chapters VI，1，2） on the development of the modular functions， insofar as the cited chapter provides a self-contained summary of Riemann's theory of algebraic functions. That， in the continued development of the modular functions， only a part of the results of the general theory are revealed， should not hinder the reader from leaving the path which led to those results. But I do not doubt， that a reader， who has only a sense of the existence theorem， and the character of the functions φ， as well as， finally， a bit of the curve-theoretic presentation methods， will have little difficulty in parts III，IV，V A full understanding of the general theory of algebraic correspondences and the modular correspondences requires a somewhat more thoroughgoing overview of the theory of algebraic functions and requires a more penetrating study of the chapters in question. The applicability of the general Riemann existence theorems to the fundamental function-theoretic problem of the modular theory leads， in chapter III，3， to the general solution of this problem， and one must consider， in this sense， chapter III，3 as the center of this entire work， The four following chapters of the third section， as well as the last chapters of the sec- one section， represent detailed investigations， and are to be considered a part of thegeneral theory III，3. That hereby， the thoroughly interesting theory of the seventh level ascended to a prominent position， is because of the intention， that the corresponding