SECT. V.] FINITE EXPRESSION OF THE SOLUTION. 167 



If in equation (B) we denote the first term of the second mem 

 ber by p and the second by q, we have 



, N tan p -f tan a 2e~ x cos y 2 cos y 

 whence tan (p + g) or - f - - == - - txf = - ^ ; 



1 tan p tan q 1 e e e 



1 /2 cos y\ .f 



whence we deduce the equation -TTV = arc tan ( -- - _-} ...(..(G). 



A \& e J 



This is the simplest form under which the solution of the 

 problem can be presented. 



206. This value of v or c/&amp;gt; (x, y) satisfies the conditions relative 

 to the ends of the solid, namely, (/&amp;gt; (x, JTT) = 0, and (j&amp;gt; (0, y} = 1 ; 



70 72 



it satisfies also the general equation + - 2 = 0, since equa 



tion ((7) is a transformation of equation (B). Hence it represents 

 exactly the system of permanent temperatures ; and since that 

 state is unique, it is impossible that there should be any other 

 solution, either more general or more restricted. 



The equation (C) furnishes, by means of tables, the value of 

 one of the three unknowns v, x, y } when two of them are given; it 

 very clearly indicates the nature of the surface whose vertical 

 ordinate is the permanent temperature of a given point of the 

 solid plate. Finally, we deduce from the same equation the values 



of the differential coefficients -=- and -y- which measure the velo- 



ax ay 



city with which heat flows in the two orthogonal directions ; and 

 we consequently know the value of the flow in any other direction. 



These coefficients are expressed thus, 



dx 

 dv 



It may be remarked that in Article 194 the value of -j- , and 

 that of -j- are given by infinite series, whose sums may be easily 



