246 THEORY OF HEAT. [CHAP. TV. 



the cosine of any multiple whatever of W A is equal to the cosine 

 of the same multiple of w A , and that the sine of any multiple 

 of % differs only in sign from the sine of the same multiple 

 of UK. It follows from this that when we unite into one the 

 two corresponding terms of each of the equations, the two un 

 knowns A^ and A A , which enter into these equations, are replaced 

 by a single unknown, namely A^ A^. As to the two unknown 

 B^ and BX they also are replaced by a single one, namely J5 A + BX : 

 it follows from this that the number of unknowns is equal in all 

 cases to the number of equations ; for the number of terms is 

 always i + 1. We must add that the unknown A disappears of 

 itself from the first terms, since it is multiplied by the sine of 

 a nul arc. Further, when the number n is even, there is found 

 at the end of each equation a term in which one of the unknowns 

 disappears of itself, since it multiplies a nul sine ; thus the 

 number of unknowns which enter into the equations is equal 

 to 2 (i + 1) 2, when the number n is even ; consequently the 

 number of unknowns is the same in all these cases as the number 

 of equations. 



266. To express the general values of the temperatures 

 a i&amp;gt; a 2 &amp;gt; a s &quot; a n&amp;gt; tne fc&amp;gt; re g m g analysis furnishes us with the equa 

 tions 



/ . 2-f A 27T\ -* verBinO 2 -? 



a = [A. sin 0.0 H-^ cos 0.0 }e m 

 1 \ n n / 



f A 1 27r D i 2lT\ 



+ M 9 sm0.1 +_B 2 cos0.1 }e 



\ n n J 



sin . 2 + B cos . 2 ~ e 



n n 



&c., 



