198 THEORY OF HEAT. [CHAP. III. 



This convergent series gives always the value of the ordinate 

 z or the distance of any point whatever of the surface from the 

 plane of x and y. 



The series formed of sines or cosines of multiple arcs are 

 therefore adapted to represent, between definite limits, all possible 

 functions, and the ordinates of lines or surfaces whose form is 

 discontinuous. Not only has the possibility of these develop 

 ments been demonstrated, but it is easy to calculate the terms 

 of the series; the value of any coefficient whatever in the 

 equation 



&amp;lt;j) (x) = a x sin x -f &amp;lt;3 2 sin 2# + a 3 sin 3# + . . . -f a t sin ix + etc., 

 is that of a definite integral, namely, 



2 



- \d&amp;gt; (as) sin i 

 TT J 



ix dx. 



Whatever be the function &amp;lt; (x), or the form of the curve 

 which it represents, the integral has a definite value which may 

 be introduced into the formula. The values of these definite 



integrals are analogous to that of the whole area I (/&amp;gt; (x) dx in 

 cluded between the curve and the axis in a given interval, or to 

 the values of mechanical quantities, such as the ordinates of the 

 centre of gravity of this area or of any solid whatever. It is 

 evident that all these quantities have assignable values, whether 

 the figure of the bodies be regular, or whether we give to them 

 an entirely arbitrary form. 



230. If we apply these principles to the problem of the motion 

 of vibrating strings, we can solve difficulties which first appeared 

 in the researches of Daniel Bernoulli. The solution given by this 

 geometrician assumes that any function whatever may always be 

 developed in a series of sines or cosines of multiple arcs. Now 

 the most complete of all the proofs of this proposition is that 

 which consists in actually resolving a given function into such a 

 series with determined coefficients. 



In researches to which partial differential equations are ap 

 plied, it is often easy to find solutions whose sum composes a 

 more general integral ; but the employment of these integrals 

 requires us to determine their extent, and to be able to dis- 



