SECT. IV.] TRANSFORMATION OF FUNCTIONS. 433 



The trigonometrical series equal to the second member is 

 convergent; the meaning of this statement is, that if we give to 

 the variable x any value whatever, the sum of the terms of the 

 series approaches more and more, and infinitely near to, a definite 

 limit. This limit is 0, if we have substituted for x a quantity 

 included between and X, but not included between a and ft; 

 but if the quantity substituted for x is included between a and b, 

 the limit of the series has the same value as f(x). The last 

 function is subject to no condition, and the line whose ordinate it 

 represents may have any form; for example, that of a contour 

 formed of a series of straight lines and curved lines. We see by 

 this that the limits a and b, the w^hole interval X, and the nature 

 of the function being arbitrary, the proposition has a very exten 

 sive signification ; and, as it not only expresses an analytical 

 property, but leads also to the solution of several important 

 problems in nature, it w r as necessary to consider it under different 

 points of view, and to indicate its chief applications. We have 

 given several proofs of this theorem in the course of this work. 

 That which we shall refer to in one of the following Articles 

 (Art. 424) has the advantage of being applicable also to non- 

 periodic functions. 



If we suppose the interval X infinite, the terms of the series 

 become differential quantities ; the sum indicated by the sign 2 

 becomes a definite integral, as was seen in Arts. 353 and 355, and 

 equation (A) is transformed into equation (B). Thus the latter 

 equation (B) is contained in the former, and belongs to the case 

 in which the interval X is infinite: the limits a and b are then 

 evidently entirely arbitrary constants. 



419. The theorem expressed by equation (B) presents also 

 divers analytical applications, which we could not unfold without 

 quitting the object of this work; but we will enunciate the 

 principle from which these applications are derived. 



We see that, in the second member of the equation 



the function f(x) is so transformed, that the symbol of the 



function / affects no longer the variable &, but an auxiliary 



F. H. 28 



