244 THEORY OF HEAT. [CHAP. IV. 



264. The equations of Art. 260 could equally be satisfied by 

 constructing the values of each one of the variables a x , a a , 8 , ... a n 

 out of the sum of the several particular values which have been 

 found for that variable ; and each one of the terms which enter 

 into the general value of one of the variables may also be mul 

 tiplied by any constant coefficient. It follows from this that, 

 denoting by A v B I} A 2 , B 2 , A 3 , B s , ...*A n) B n) any coefficients 

 whatever, we may take to express the general value of one of the 

 variables, a^j for example, the equation 



/ r&amp;gt; \ ^n vers i n M i 



of wi+l == (A i sin mu l 4 B^ cos muj e 



versin 11% 



+ (A* sin mu&amp;gt;, 4- B cos mu) e &quot; 



-?** versinw,, 



+ (A n sin mu n + B n cos mu n ) e 7&amp;lt; 



The quantities A lt A^A 33 ... A n , J5 X , J5 a , J5 8 , ... B n , which 

 enter into this equation, are arbitrary, and the arcs u it u 2 ,u s , ... u n 

 are given by the equations : 



A 2?r - 2?r 2?r 27T 



^ = 0-, ^ 2 = 1-, &quot;. = 2-, ..., Wn =(^l)-. 



The general values of the variables cfj, a a , a 8 , ... a n are then 

 expressed by the following equations : 



. _ 



a t = (A l sin Ow t + B l cos OuJ e 5 



sn w + cos 



_ versin 3 



sn w + cos * 



&c.; 



_m versin 



2 = (A^ sin lu^ + B^ cos IttJ e 



- versin w 2 



4 (A 2 sin \u z 4- B 2 cos Iw 2 ) e 1 



~ versin % 



+ (A a sin lu s 4 B 3 cos lnj e 

 + &c.; 



