SECT. II.] DIFFERENT INTEGRAL FORMS. 2G7 



and constitute no different analysis. In oneofthe following 

 chapters we shall examine the different forms whicfT may be 



assumed by the integral of the equation -r ^-r^^ the relations 



dv dx 



which they have to each other, and the cases in which they ought 

 to be employed. 



To form the integral which expresses the movement of heat in 

 a ring, it was necessary to resolve an arbitrary function into a 

 series of sines and cosines of multiple arcs; the numbers which 

 affect the variable under the symbols sine and cosine are the 

 natural numbers 1, 2, 3, 4, &c. In the following problem the 

 arbitrary function is again reduced to a series of sines; but the 

 coefficients of the variable under the symbol sine are no longer 

 the numbers 1, 2, 3, 4, &c.: these coefficients satisfy a definite 

 equation whose roots are all incommensurable and infinite in 

 number. 



Note on Sect. I, Chap. IV. Guglielmo Libri of Florence was the first to 

 investigate the problem of the movement of heat in a ring on the hypothesis of 

 the law of cooling established by Dulong and Petit. See his Memoire sur la 

 theorie de la chaleur, Crelle s Journal, Band VII., pp. 116131, Berlin, 1831. 

 (Read before the French Academy of Sciences, 1825. ) M. Libri made the solution 

 depend upon a series of partial differential equations, treating them as if they 

 were linear. The equations have been discussed in a different manner by 

 Mr Kelland, in his Theory of Heat, pp. 69 75, Cambridge, 1837. The principal 

 result obtained is that the mean of the temperatures at opposite ends of any 

 diameter of the ring is the same at the same instant. [A. F.] 



