290 THEORY OF HEAT. {CHAP. V. 



and consider cos e as derived from sin e by differentiation. 



Suppose that instead of forming sin e from the product of an 

 infinite number of factors, we employ only the m first, and denote 

 the product by &amp;lt;/&amp;gt; w ( 6 )* To find the corresponding value of cose, 

 we take 



*. or $ (). 



This done, we have the equation 



*.W-*. () = o. 



Now, giving to the number m its successive values 1, 2, 3, 4, &a 

 from 1 to infinity, we ascertain by the ordinary principles of 

 Algebra, the nature of the functions of e which correspond to 

 these different values of m. We see that, whatever m the number 

 of factors may be, the equations in e which proceed from them 

 have the distinctive character of equations all of whose roots 

 are real. Hence we conclude rigorously that the equation 



in which X is less than unity, cannot have an imaginary root 1 . 

 The same proposition could also be deduced by a different analysis 

 which we shall employ in one of the following chapters. 



Moreover the solution we have given is not founded on the 

 property which the equation possesses of having all its roots 

 real. It would not therefore have been necessary to prove 

 this proposition by the principles of algebraical analysis. It 

 is sufficient for the accuracy of the solution that the integral 

 can be made to coincide with any initial state whatever; for 

 it follows rigorously that it must then also represent all the 

 subsequent states. 



1 The proof given by Eiemann, Part. Diff. Gleich. 67, is more simple. The 

 method of proof is in part claimed by Poisson, Bulletin de la Societe Philomatique, 

 Paris, 1826, p. 147. [A. F.]. 



