352 THEORY OF HEAT. [CHAP. IX. 



stated in Article 2:34, which gives the development of any func 

 tion F(x) in a series of sines and cosines of multiple arcs. We 

 pass from the last proposition to those which we have just demon 

 strated, by giving an infinite value to the dimensions. Each term 

 of the series becomes in this case a differential quantity 1 . Trans 

 formations of functions into trigonometrical series are some of the 

 elements of the analytical theory of heat; it is indispensable to 

 make use of them to solve the problems which depend on this 

 theory. 



The reduction of arbitrary functions into definite integrals, 

 such as are expressed by equation (E), and the two elementary 

 equations from which it is derived, give rise to different conse 

 quences which are omitted here since they have a less direct rela 

 tion with the physical problem. We shall only remark that the 

 same equations present themselves sometimes in analysis under 

 other forms. We obtain for example this result 



1 r r 

 &amp;lt;j&amp;gt;(x)=- drf (a) I dqcosq(x a) (E f ) 



TfJ JO 



which differs from equation (E) in that the limits taken with 

 respect to a are and oo instead of being oo and + oo . 



In this case it must be remarked that the two equations (E) 

 and (E ) give equal values for the second member when the 

 variable x is positive. If this variable is negative, equation (E 1 ) 

 always gives a nul value for the second member. The same is 

 not the case with equation (E), whose second member is equiva 

 lent to 7T(j) (x), whether we give to x a positive or negative value. 

 As to equation (E ) it solves the following problem. To find a 

 function of x such that if x is positive, the value of the function 

 may be &amp;lt;/&amp;gt; (x), and if x is negative the value of the function may 

 be always nothing 2 . 



363. The problem of the propagation of heat in an infinite 

 line may besides be solved by giving to the integral of the partial 

 differential equation a different form which we shall indicate in 



1 Eiemann, Part. Diff. Gleich. 32, gives the proof, and deduces the formulae 

 corresponding to the cases F (x) = F ( - x). 



2 These remarks are essential to clearness of view. The equations from which 

 (E) and its cognate form may be derived will be found in Todhunter s Integral 

 Calculus, Cambridge, 1862, 316, Equations (3) and (4). [A. F.] 



