206 THEORY OF HEAT. [CHAP. III. 



x}dx (II) 



27T03 f-, . ZTTX , 4-TnE f .. , 47nc , , 



-f cos -yr- I / (x) cos -TF- a# + cos =^ I /(a?) cos -^- a# 4- &c. 



. ZTTX f /. / N . 27HB 7 . 4urx [ - , . . 47r# , p 

 -f sin TT- / (a?) sin TT- a# + sin - v Ifw sin -^- a^ + &c. 

 JL J .A- -A- J & 



235. It follows from that which has been proved in this sec 

 tion, concerning the development of functions in trigonometrical 

 series, that if a function f(x) be proposed, whose value in a de 

 finite interval from x = to x = X is represented by the ordinate 

 of a curved line arbitrarily drawn ; we can always develope this 

 function in a series which contains only sines or only cosines, or 

 the sines and cosines of multiple arcs, or the cosines only of odd 

 multiples. To ascertain the terms of these series we must employ 

 equations (M), (N), (P). 



The fundamental problems of the theory of heat cannot be 

 completely solved, without reducing to this form the functions 

 which represent the initial state of the temperatures. 



These trigonometric series, arranged according to cosines or 

 sines of multiples of arcs, belong to elementary analysis, like the 

 series whose terms contain the successive powers of the variable. 

 The coefficients of the trigonometric series are definite areas, and 

 those of the series of powers are functions given by differentiation, 

 in which, moreover, we assign to the variable a definite value. We 

 could have added several remarks concerning the use and pro 

 perties of trigonometrical series ; but we shall limit ourselves to 

 enunciating briefly those which have the most direct relation to 

 the theory with which we are concerned. 



it follows that 



f(x) =A + AI cos 1- ^ 2 cos 2 h^ 3 cos 3 + &c. 



A A A 



. 2-7TX _ STTX ITTX 



+ B^ sin -r + B 2 sin 2 - + B% sin 3 -r + &c. 

 A A A 



The coefficients being determined in Fourier s manner by multiplying both 



sides by . n . - and integrating from to X. (Philosophical Magazine, August 

 sin A 



1874, pp. 95, 9G). [A. F.j 



