208 THEORY OF HEAT. [CHAP. III. 



or 



F(x) = - I F(-J) doi Ji + cos (x - a) + cos 2 (x - a) + &c. j . 



Hence, denoting the sum of the preceding series by 



2 cos i (x a) 

 taken from i = 1 to i = GO , we have 



F(x)=- \F (a) d* \l + S cos i(x - a)! . 



7TJ [Z J 



The expression ^ + X cos i (a? a) represents a function of # 



2 



and a, such that if it be multiplied by any function whatever F(oi), 

 and integrated with respect to a between the limits a = TT and 

 a = ?r, the proposed function jP(a) becomes changed into a like 

 function of x multiplied by the semi-circumference TT. It will be 

 seen in the sequel what is the nature of the quantities, such as 



5 + 2cos*(# a), which enjoy the property we have just enun- 



2 



ciated. 



4th. If in the equations (M), (N), and (P) (Art 234), which 

 on being divided by r give the development of a function f(x), 

 we suppose the interval r to become infinitely large, each term of 

 the series is an infinitely smal^ element of an integral; the sum of 

 the series is then represented by a definite integral. When the 

 bodies have determinate dimensions, the arbitrary functions which 

 represent the initial temperatures, and which enter into the in 

 tegrals of the partial differential equations, ought to be developed 

 in series analogous to those of the equations (M), (N), (P) ; but 

 \ these functions take the form of definite integrals, when the 

 dimensTons of the bodies are not determinate, as will be ex 

 plained in the course of this work, in treating of the free diffusion 

 of heat (Chapter IX.). 



Note on Section VI. On the subject of the development of a function whose 

 values are arbitrarily assigned between certain limits, in series of sines and 

 cosines of multiple arcs, and on questions connected with the values of such 

 series at the limits, on the convergency of the series, and on the discontinuity 

 of their values, the principal authorities are 



Poisson. Theorie mathematiqiie de la Chaleur, Paris, 1835, Chap. vn. Arts. 

 92 102, Sur la maniere d exprimcr les fonctions arbitraircs par des series de 



