134 THEORY OF HEAT. [CHAP. III. 



belongs to the actual problem, and determines the properties 

 of the required curved surface, is the following : 



The function of a? and y&amp;gt; &amp;lt;f&amp;gt; (x, y), which represents the per 

 manent state of the solid BA G, must, 1st, satisfy the equation 

 (a) ; 2nd, become nothing when we substitute J TT or + \ir for y, 

 whatever the value of x may be ; 3rd, must be equal to unity 

 when we suppose x = and y to have any value included between 

 J TT and + i TT. 



Further, this function &amp;lt; (x, y) ought to become extremely 

 small when we give to x a very large value, since all the heat 

 proceeds from the source A. 



167. In order to consider the problem in its elements, we 

 shall in the first place seek for the simplest functions of x 

 and y, which satisfy equation (a) ; we shaTT then generalise the 

 value of v in order to satisfy all the stated conditions. By this 

 method the solution will receive all possible extension, and we 

 shall prove that the problem proposed admits of no other 

 solution. 



Functions of two variables often reduce to less complex ex 

 pressions, when we attribute to one of the variables or to both 

 of them infinite values ; this is what may be remarked in alge 

 braic functions which, in this particular case, take the form of 

 the product of a function of x by a function of y. 



We shall examine first if the value of v can be represented 

 by such a product ; for the function v must represent the state 

 of the plate throughout its whole extent, and consequently that 

 of the points whose co-ordinate x is infinite. We shall then 

 write v = F(x)f(y}\ substituting in equation (a) and denoting 



by F&quot; (x) and by/ (y\ we shall have 



(*) ,/ (y)_ . 







we then suppose \^ = m and r^ = m&amp;gt;&amp;gt; m being any 



