298 THEORY OF HEAT. [CHAP. VI. 



with the equation 



2 cos (a sin x) = A 4- B cos Zx + C cos 4# + &c., 



we shall find the values of the coefficients A, B, G expressed by 

 definite integrals. It is sufficient here to find that of the first 

 coefficient A. We have then 



- A = - I cos (a sin x) dx, 



the integral should be taken from x = to x = TT. Hence the 

 value of the series 1 ^ + ^ T* ~ w~4? 6* + ^ c&amp;gt; * s ^ iat ^ tne 



definite integral dx cos (a sin x). We should find in the same 



Jo 



manner by comparison of two equations the values of the successive 

 coefficients B, G, &c.; we have indicated these results because they 

 are useful in other researches which depend on the same theory. 

 It follows from this that the particular value of u which satisfies 

 the equation 



d*u Idu .If , /- . . 7 

 9 U + j + ~ c = 1S -J cos ( ^ sm *) fo* 



the integral being taken from r = to r = TT. Denoting by q this 



[dx 



value of u, and making u = qS, we find S = a + & 2 &amp;gt; an d we have 



J #2 



as the complete integral of the equation gu + ^ 2 + - -r- = 0, 



u == | a 4-6 | T? -- &amp;gt;2 /cos (a; ^ sin r) Jr. 

 j a? in r J j 



&amp;gt; 

 ] Jcos (asjg sin r) dr\ 



a and & are arbitrary constants. If we suppose 6 = 0, we have, 

 as formerly, 



u = I cos (x Jg sin r) dr. 



With respect to this expression we add the following remarks. 



^^ _ uijmjjuBjjpinwr 







311. The equation 



If&quot; /9 2 /9* /9 6 



- J cos (^ sin w) c? M = 1 - ^ + ^-g - gi-pTgi + &c. 



