PREFACE TO THE SECOND EDITION. WHEN I commenced to prepare for the press a Second Edition of the late Dr Boole's Treatise on Finite Differences, my intention was to leave the work unchanged save by the insertion of sundry additions in the shape of paragraphs marked off from the rest of the text. But I soon found that adherence to such a principle would greatly lessen the value of the book as a Text-book, since it would be impossible to avoid confused arrangement and even much repetition. I have therefore allowed myself considerable freedom as regards the form and arrangement of those parts where the additions are considerable, but I have strictly adhered to the principle of inserting all that was contained in the First Edition. As such Treatises as the present are in close connexion with the course of Mathematical Study at the University of Cambridge, there is considerable difficulty in deciding the question how far they should aim at being exhaustive. I have held it best not to insert investigations that involve complicated analysis unless they possess great suggestiveness or are the bases of important developments of the subject. Under the present system the premium on wide superficial reading is so great that such investigations, if inserted, would seldom be read. But though this is at present the case, there is every reason to hope that it will not continue to be and in view of a time when students will aim at an exhaustive study of a few subjects in preference to a superficial acquaintance with the whole range of Mathematical research, I have added brief notes referring to most of the papers on the subjects of this Treatise that have appeared in the Mathematical Serials, and to other original sources. In virtue of such references, and the brief indication of the subject of the paper that accompanies each, it is hoped that this work may serve as a handbook to students who wish to read the subject more thoroughly than they could do by confining themselves to an Educational Text-book. The latter part of the book has been left untouched. Much of it I hold to be unsuited to a work like the present, partly for reasons similar to those given above, and partly because it treats in a brief and necessarily imperfect manner subjects that had better be left to separate treatises. It is impossible within the limits of the present work to treat adequately the Calculus of Operations and the Calculus of Functions, and I should have preferred leaving them wholly to such treatises as those of Lagrange, Babbage, Carmichael, De Morgan, &c. I have therefore abstained from making any additions to these portions of the book, and have made it my chief aim to render more evident the remarkable analogy between the Calculus of Finite Differences and the Differential Calculus. With this view I have suffered myself to digress into the subject of the Singular Solutions of Differential Equations, to a much greater extent than Dr Boole had done. But I trust that the advantage of rendering the investigation a complete one will be held to justify the irrelevance of much of it to that which is nominally the subject of the book. It is partly from similar considerations that I have adopted a nomenclature slightly differing from that commonly used (e. g. Partial Difference-Equations for Equations of Partial Differences). I am greatly indebted to Mr R. T. Wright of Christ's College for his kind assistance. He has revised the proofs for me, and throughout the work has given me valuable suggestions of which I have made free use. JOHN F. MOULTON. CHRIST'S COLLEGE, Oct. 1872. Differences of Elementary Functions, 6. Expansion in factorials, 11. Generating Functions, 14. Laws and relations of E, A and ON INTERPOLATION, AND MECHANICAL QUADRATURE . Nature of the Problem, 33. Given values equidistant, 34. Not equi- distant-Lagrange's Method, 38. Gauss' Method, 42. Cauchy's Method, 43. Application to Statistics, 43. Areas of Curves, 46. Weddle's rule, 48. Gauss' Theorem on the best position of the given ordinates, 51. Laplace's method of Quadratures, 53. Refer- ences on Interpolation, &c. 55. Connexion between Gauss' Theo- rem and Laplace's Coefficients, 57. Exercises, 57. FINITE INTEGRATION, AND THE SUMMATION OF SERIES Meaning of Integration, 62. Nature of the constant of Integration, Rational and integral Functions, 68. Integrable Fractions, 70. Functions of the form a«p(x), 72. Miscellaneous Forms, 75. Repeated Integration, 77. Conditions of extension of direct to inverse forms, 78. Periodical constants, 80. Analogy between the Integral and Sum-Calculus, 81. References, 83. Exercises, Dévelopment of 2, 87. Analogy with the methods adopted for the development of s, 87 (note). Division of the problem, 88. De- velopment of E in powers of D (Euler-Maclaurin Formula), 89. Values of Bernoulli's Numbers, 90. Applications, 91. Deter- mination of Constant, 95. Development of 2, 96.: Development of Euz and Enuz in differences of a factor of Uz, 99. Method of in. creasing the degree of approximation obtained by Maclaurin Theorem, 100. Expansion in inverse factorials, 102. References, BERNOULLI'S NUMBERS, AND FACTORIAL COEFFICIENTS . Various expressions for Bernoulli’s Numbers—De Moivre's, 107. In terms of Σ. 109. Raabe's (in factors), 109. As definite inte- grals, 110. Euler's Numbers, 110. Bauer's Theorem, 112. Fac- Definitions, 123. Case in which Uze has a periodic factor, 124. Cauchy's Proposition, 126. First derived Criterion, 129. Supplemental Criteria-Bertrand's Form, 132. De Morgan's Form, 134. Third Form, 135. Theory of Degree, 136. Application of Tests to the Euler-Maclaurin Formula, 139. Order of Zeros, 139. Refer- |