186 THEORY OF HEAT. [CHAP. III. 



220. We see by this that the coefficients a, b, c, d, e,f, &c., 

 which enter into the equation 



5 Tr&amp;lt;p (x) a sin x + b sin 2x + c sin 3x + d sin 4# + &c., 



and which we found formerly by way of successive eliminations, 

 are the values of definite integrals expressed by the general term 



sin ix (j&amp;gt; (x) dx } i being the number of the term whose coefficient 



is required. This remark is important, because it shews how even 

 entirely arbitrary functions may be developed in series of sines 

 of multiple arcs. In fact, if the function &amp;lt; (x) be represented 

 by the variable ordinate of any curve whatever whose abscissa 

 extends from x = to x TT, and if on the same part of the axis 

 the known trigonometric curve, whose ordinate is y sin x, be 

 constructed, it is easy to represent the value of any integral 

 term. We must suppose that for each abscissa x, to which cor 

 responds one value of $ (a?), and one value of sin x, we multiply 

 the latter value by the first, and at the same point of the axis 

 raise an ordinate equal to the product $ (x) sin x. By this con 

 tinuous operation a third curve is formed, whose ordinates are 

 ~those of the trigonometric curve, reduced in proportion to the 

 ^ordinates of the arbitary curve which represents &amp;lt;(#). This 

 done, the area of the reduced curve taken from x = to X = TT 

 gives the exact value of the coefficient of sin#; and whatever 

 the given curve may be which corresponds to $ (#), whether we 

 can assign to it an analytical equation, or whether it depends on 

 110 regular law, it is evident that it always serves to reduce 

 in any manner whatever the trigonometric curve; so that the 

 area of the reduced curve has, in all possible cases, a definite 

 value, which is the value of the coefficient of sin x in the develop 

 ment of the function. The same is the case with the following 



coefficient b, or /&amp;lt; (x) sin 2xdx. 



In general, to construct the values of the coefficients a, b, c, d, &c., 

 \\e must imagine that the curves, whose equations are 



y = sin x, y = sin Zx, y = sin Sx, y = sin 4#, &c., 

 have been traced for the same interval on the axis of x, from 



