Contents. 



Page 



Introductiou 1 



Part I. Continuüus Theory. 



Art. 



1 Formulatioii of the problem 7 



2 — 5 Analytical representation of the conditions of the phenomenon 7 



6 — 10 Solutions of Fourier's type 9 



11—18 Behaviour of F' (a:, for t small 11 



19 Summary relating to the behaviour of V {x, t) for t small 20 



20—21 Behaviour of V{x,t) for t small 20 



22 Necessary and sufficient conditions for f(z) 21 



23 Stahle, unstable, and inadmissible initial states 22 



24 Illustrative examples 22 



Part II. Discontinuous Theory. 



25 Formulation of a discontinuous theory of solids 25 



26 — 31 Approximate analytical representation of the conditions of the phenomenon .... 26 



32 Auxiliary functions of Fourier's type 33 



33 — 36 Superior limits of E, b, and e 34 



37 — 38 Sufficient and quasi-necessary conditions ior f{x) 37 



39 Illustrative examples 39 



40 — 42 Criticism of Fourier's theory 42 



43 Illustrative examples 44 



Part III. Improperly Continuous Theory. 



44 Notation of improperly continuous analysis 48 



45 Formulation of an improperly continuous theory of solids 49 



46 — 48 Analytical representation of the conditions of the phenomenon 51 



49 — 51 Characteristic functions of Fourier's type 52 



52 Necessary and sufficient conditions for cp Ü) 54 



53 Stahle, unstable, and inadmissible initial states 55 



54 — 55 Approximations to impossible initial states 55 



56 Illustrative examples 57 



57 — 59 Uniqueness of the Solution 61 



Part IV. Summary. 



60 — 61 The relation of mathematical analysis to physics 65 



62 The object of the essay 65 



63 The continuous theory 66 



64 The discontinuous theory. . 67 



65 The improperly continuous theory 67 



12 JAN. 1904 



