CHAP. VII.] DETERMINATION OF THE COEFFICIENTS. 313 



Denoting by 1 the constant temperature of the end A, assume 

 the two equations 



1 = a : cos njj + 2 cos n z y + a 3 cos ?? z y + &c , 

 1 = \ cos n$ + b 2 cos v 2 y + & 3 cos njj + &c. 



It is sufficient then to determine the coefficients a lf a a , a- 3 , &c., 

 whose number is infinite, so that the second member of the equa 

 tion may be always equal to unity. This problem has already 

 been solved in the case where the numbers n lt n 3 , n s , &c. form the 

 series of odd numbers (Chap. III., Sec. IL, Art. 177). Here 

 ?ij, n 2&amp;gt; n 3 j &c. are incommensurable quantities given by an equa 

 tion of infinitely high degree. 



324. Writing down the equation 



1 = dj cos n^y + a a cos n$ + a 3 cos n. A y + &c., 



multiply the &quot;two members of the equation by cos n^y dy, and take 

 the integral from y = to y l. We thus determine the first 

 coefficient a r The remaining coefficients may be determined in a 

 similar manner. 



In general, if we multiply the two members of the equation by 

 cos vy, and integrate it, we have corresponding to a single term 

 of the second member, represented by a cos ny t the integral 



a Icos ny cos vy dij or ^al cos (n v) y dy + -^ a /cos (n + v) ydy, 



sin (n &quot; &quot;)* + ^T V sin (n +v] 



and making y=-l t 



a ((n 4- ii) sin (n v)l+(n v) sin (n -f- v)J.\ 



a I -~tf~?~ y 



Now, every value of w satisfies the equation wtanw/ = T; the 

 same is the case with v, we have therefore 

 n tan vl = v tan z^Z ; 

 or n sin w cos vl v sin i/ cos ?z = 0. 



