300 THEORY OF HEAT. [CHAP. VI. 



Now, it is easy to see that the values of 8 lt $ 3 , $ 5 , &c. are 

 nothing. With respect to $ 2 , $ 4 , S R) &c. their values are the 

 quantities which we previously denoted by A# A# A R , &c. For 

 this reason, substituting these values in the equation (e) we have 

 generally, whatever the function &amp;lt;/&amp;gt; may be, 





u) du 



in the case in question, the function $ (z) represents cos z, and we 

 have (j&amp;gt; = 1, &amp;lt;/&amp;gt;&quot; = 1, &amp;lt; iv = 1, &amp;lt;/&amp;gt;* = 1, and so on. 



312. To ascertain completely the nature of the function / (0), 

 and of the equation which gives the values of g, it would be 

 necessary to consider the form of the line whose equation is 



which forms with the axis of abscissae areas alternately positive 

 and negative which cancel each other ; the preceding remarks, also, 

 on the expression of the values of series by means of definite 

 integrals, might be made more general. When a function of the 

 variable x is developed according to powers of x, it is easy to 

 deduce the function which would represent the same series, if the 

 powers x, x*, x 3 , &c. were replaced by cos x, cos 2aj, cos 3x, &c. By 

 making use of this reduction and of the process employed in the 

 , second paragraph of Article 235, we obtain the definite integrals 

 which are equivalent to given series ; but we could not enter upon 

 this investigation, without departing too far from our main object. 



It is sufficient to have indicated the methods which have 

 enabled us to express the values of series by definite integrals. 



We will add only the development of the quantity 6 fj^ in a 

 continued fraction. 



313. The undetermined y orf(0) satisfies the equation 



