SECT. I.] INTRODUCTION. 21 



has taught us. The problem is reducible to the hypothesis that 

 every point of a vast sphere is affected by periodic temperatures ; 

 analysis then tells us according to what law the intensity of these 

 variations decreases according as the depth increases, what is the 

 amount of the annual or diurnal changes at a given depth, the 

 epoch of the changes, and how the fixed value of the underground 

 temperature is deduced from the variable temperatures observed 

 at the surface. 



13. The general equations of the propagation of heat are 

 partial differential equations, and though their form is very simple 

 the known methods l do not furnish any general mode of integrat 

 ing them; we could not therefore deduce from them the values 

 of the temperatures after a definite time. The numerical inter 

 pretation of the results of analysis is however necessary, and it 

 is a degree of perfection which it would be very important to give 

 to every application of analysis to the natural sciences. So long 

 as it is not obtained, the solutions may be said to remain in 

 complete and useless, and the truth which it is proposed to 

 discover is no less hidden in the formulas of analysis than it was 

 in the physical problem itself. We have applied ourselves with 

 much care to this purpose, and we have been able to overcome 

 the difficulty in all the problems of which we have treated, and 

 which contain the chief elements of the theory of heat. There is 

 not one of the problems whose solution does not provide conve 

 nient and exact means for discovering the numerical values of the 

 temperatures acquired, or those of the quantities of heat which 



1 For the modern treatment of these equations consult 



Partielle Differentialgleichungen, von B. Eiemann, Braunschweig, 2nd Ed., 1876. 

 The fourth section, Bewegung der Warme in festen Korpern. 



Cours de physique mathematique, par E. Matthieu, Paris, 1873. The parts 

 relative to the differential equations of the theory of heat. 



The Functions of Laplace, Lame, and Bessel, by I. Todhunter, London, 1875. 

 Chapters XXI. XXV. XXIX. which give some of Lame s methods. 



Conferences de Physique, par E. Verdet, Paris, 1872 [(Euvres, Vol. iv. Part i.]. 

 Legons sur la propagation de la chaleur par conductibilite. These are followed by 

 a very extensive bibliography of the whole subject of conduction of heat. 



For an interesting sketch and application of Fourier s Theory see 



Theory of Heat, by Prof. Maxwell, London, 1875 [4th Edition]. Chapter XVIII. 

 On the diffusion of heat by conduction. 



Natural Philosophy, by Sir W. Thomson and Prof. Tait, Vol. i. Oxford, 1867. 

 Chapter VII. Appendix D, On the secular cooling of the earth. [A. F. ] 



