210 THEORY OF HEAT. [CHAP. ITT. 



Suppose the initial temperature at all points of the slab BAG 

 to be nothing, but that the temperature at each point in of the 

 edge A is preserved by some external cause, and that its fixed 

 value is a function f(x) of the distance of the point m from the 

 end of the edge A whose whole length is 2r; let v be the 

 constant temperature of the point m whose co-ordinates are x and 

 y, it is required to determine v as a function of x and y. 



The value v = ae~ mv sin mx satisfies the equation 



HT 



a and m being any quantities whatever. If we take m = i - , 



i being an integer, the value ae *&quot; r sin vanishes, when x = r, 



whatever the value of y may be. We shall therefore assume, as a 

 more general value of v, 



. - . - . 



v = a,e r sin -- \- a t&amp;gt; e r sin - + ae r sin -- h &c. 

 r r r 



If y be supposed nothing, the value of v will by hypothesis 

 be equal to the known function f(x). We then have 



/., x . . . 



j (x) = a^ sin + & 2 sin -- \- a a sin -- f- &c. 



The coefficients a lt a 2 , 3 , &c. can be determined by means of 

 equation (M), and on substituting them in the value of v we have 



1 -IT- . TTX ^ ,, N . 7TX , -2ir^ . %irX C -, , . 27T# 7 



s rv = e r sm / /(a?) sm a^ + e f sin - f () sm -- dx 



2 r- 7 r r J \ i T 



o&quot; . V IM/ I // A * *V j| .0 



+ e r sin / f (x) sin dx + &c. 



237. Assuming r = TT in the preceding equation, we have the 

 solution under a more simple form, namely 



- jrv e^ sin x\f(x] sin #&amp;lt;& + e~ 2y sin 2# !./(#) sin Zxdx 



+ e~ 5v sin 3^7 / f(x\ sin 3a?c?^ + &c /. (a 



J- 7 w 



