Mr Wilson, On Convection of Heat. 419 



Then, when x is negative, we have 



0=2 A n e {p+ ^ + * )x sin ny, 



M = l 



and, when x is positive, 



0=X B n e {p -' Jn2+p ' 2)x smny. 

 Putting x = 0, we see that A n = B n . 

 Also when x = 0, 



when y is between — and -r- , and = when y is between and 



7T 37T , 



7 or -T- and 7r. 



4 4 



J3ut 



dx( x= _) dx( x=+) 



= A; (2^ m (p + J'rt+pf) sin ny — % A n (p — Jri> + p") sin ny) 



= k%A n 2 Jn 2 +p 2 sin ny. 



To determine the A's therefore we require the coefficients 

 in the series 



(f> (x) = a x sin x + a 2 sin 2x + ..., 

 where <f> (x) = from to — , 



n 7T 37T 



= Q » i t0 t ' 



t 



= „ -;- tO 7T. 



4 



3tt 



2 /""■ . 20 f 4 



We have a w = - 6 (x) sin n^cfe = — - / sin nxdx 



TTJi) 7T J „ 



4 



= 0, when n is even, 



2 V2 

 = , when n is 1, 7, 9, 15, 17, etc., 



TTll 



2V2 , 



= , when w is 3, 5, 11, 13, etc. 



7ra 



