452 THEORY OF HEAT. [CHAP. IX. 



3rd. We can prove by the fundamental theorems of algebra, 

 or even by the physical nature of the problem, that the transcen 

 dental equation has all its roots real, in number infinite. 



4th. In elementary problems, the general term takes the form 

 of a sine or cosine ; the roots of the definite equation are either 

 whole numbers, or real or irrational quantities, each of them in 

 cluded between two definite limits. 



In more complex problems, the general term takes the form of 

 a function given implicitly by means of a differential equation 

 integrable or not. However it may be, the roots of the definite 

 equation exist, they are real, infinite in number. This distinction 

 of the parts of which the integral must be composed, is very 

 important, since it shews clearly the form of the solution, and the 

 necessary relation between the coefficients. 



5th. It remains only to determine the constants which depend 

 on the initial state; which is done by elimination of the unknowns 

 from an infinite number of equations of the first degree. We 

 multiply the equation which relates to the initial state by a 

 differential factor, and integrate it between defined limits, which 

 are most commonly those of the solid in which the movement is 

 effected. 



There are problems in which we have determined the co 

 efficients by successive integrations, as may be seen in the memoir 

 whose object is the temperature of dwellings. In this case we 

 consider the exponential integrals, which belong to the initial 

 state of the infinite solid : it is easy to obtain these integrals 1 . 



It follows from the integrations that all the terms of the second 

 member disappear, except only that whose coefficient we wish to 

 determine. In the value of this coefficient, the denominator be 

 comes nul, and we always obtain a definite integral whose limits 

 are those of the solid, and one of whose factors is the arbitrary 

 function which belongs to the initial state. This form of the result 

 is necessary, since the variable movement, which is the object of 

 the problem, is compounded of all those which would have existed 

 separately, if each point of the solid had alone been heated, and 

 the temperature of every other point had been nothing. 



1 See section 11 of the sketch of this memoir, given by the author in the 

 Bulletin des Sciences par la Societe Pliilomatiqtie, 1818, pp. 111. [A. F.] 



