234 THEORY OF HEAT. [CHAP. IV. 



found in the following equations. From this. we conclude that 

 constant terms must enter into the general values of a, A 7, ... &&amp;gt;. 



Further, adding all the particular values corresponding to 

 a, /3, 7, ... &c., we have 



sin nu - verem u 



a + /3+7 + &c. = flL e r ; 



1 smu 



an equation whose second member is reduced to provided the 

 arc u does not vanish ; but in that case we should find n to be 



the value of - . We have then in general 

 sin u 



a + /3 + 7 + &c. = na l ; 



now the initial values of the variables being a, b, c, &c., we must 

 necessarily have 



na l = a + b + c + &c. ; 



it follows that the constant term which must enter into each of 

 the general values of 



a, ft, 7, ... ft) is - (a + b + c + &c.), 



that is to say, the mean of all the initial temperatures. 



As to the general values of a, A 7, ... G&amp;gt;, they are expressed 

 by the following equations : 



, Sin U Sin Ou -^ venin u* 



1 sin u 



sin u&quot; sin Ow&quot; - venm - 



+ &c., 



1 sin 2 M sin M -^vewiuu 



_ (a + & + c + &c.) + a 1 -- s -- 



Sill 2 M Sin id - versln u 



Sin 2Z*&quot; Sin u&quot; -^ vemin u&quot; 



CI - r ^ - e r 

 sin u 



&c, 



