446 THEORY OF HEAT. {CHAP. IX. 



that is to say the sum of the values /, / 2 , / 3 .../, multiplied 

 respectively by the factors 



dx sin (fjLjdx), dx sin (fjifidx], dx sin (pfidx) . . . dx sin (^ndx), 



would be reduced to zero. This relation would then necessarily 

 exist between the given quantities/, , / 2 , / 3 /; and they could not 

 be considered entirely arbitrary, contrary to hypothesis. If these 

 quantities /, f 2 ,f s ---f n have any values whatever, the relation in 

 question cannot exist, and we cannot satisfy the proposed con 

 ditions by omitting one or more terms, such as a- 3 sin (fijX) in 

 equation (e). 



Hence the function f(x) remaining undetermined, that is to 

 say, representing the system of an infinite number of arbitrary 

 constants which correspond to the values of x included between 

 and X, it is necessary to introduce into the second member of 

 equation (e) all the terms such as a. sinter), which satisfy the 

 condition 



x 



dx sin /Aft sin fi f x 0, 

 o 



the indices i and j being different; but if it happen that the 

 function /(*) is such that the n magnitudes /,/ 2 ,/ 3 -/ are 

 connected by a relation expressed by the equation 



-x 



dx sin fj,jxf(x) = 0, 

 o 



it is evident that the term c^sin/*^ might be omitted in the equa 

 tion (e). 



Thus there are several classes of functions / (x) whose develop 

 ment, represented by the second member of the equation (e), does 

 not contain certain terms corresponding to some of the roots JJL. 

 There are for example cases in which we omit all the terms 

 whose index is even; and we have seen different examples of this 

 in the course of this work. But this would not hold, if the func 

 tion /(a?) had all the generality possible. In all these cases, we 

 ought to suppose the second member of equation (e) to be com 

 plete, and the investigation shews what terms ought to be omitted, 

 since their coefficients become nothing. 



