432 THEORY OF HEAT. [CHAP. IX. 



at every point with the axis of x. This result is very remarkable, 

 and determines the true sense of the proposition expressed by 

 equation (B). 



418. The theorem expressed by equation (II) Art. 234 must 

 be considered under the same point of view. This equation 

 serves to develope an arbitrary function / (x) in a series of sines or 

 cosines of multiple arcs. The function f(x) denotes a function 

 completely arbitrary, that is to say a succession of given values, 

 subject or not to a common law, and answering to all the values of 

 x included between and any magnitude X. 



The value of this function is expressed by the following 

 equation, 



*?y(*-lO (A). 



The integral, with respect to a, must be taken between the 

 limits a = a, and a = 6 ; each of these limits a and I is any quantity 

 whatever included between and X. The sign 2 affects the 

 integer number i t and indicates that we must give to i every 

 integer value negative or positive, namely, 



...-5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5,... 



and must take the sum of the terms arranged under the sign 2. 

 After these integrations the second member becomes a function of 

 the variable x only, and of the constants a and b. The general 

 proposition consists in this : 1st, that the value of the second 

 member, which would be found on substituting for x a quantity 

 included between a and &, is equal to that which would be obtained 

 on substituting the same quantity for x in the function /(a?); 2nd, 

 every other value of x included between and X, but not included 

 between a and b, being substituted in the second member, gives a 

 mil result. 



Thus there is no function f(x), or part of a function, which 

 cannot be expressed by a trigonometric series. 



The value of the second member is periodic, and the interval 

 of the period is X, that is to say, the value of the second member 

 does not change when x + X is written instead of x. All its 

 values in succession are renewed at intervals X. 



