324* THEORY OF HEAT. [CHAP. VIII. 



These ought to be satisfied when x a, or y a, or g a, 

 The centre of the cube is taken to be the origin of co-ordinates : 

 and the side is denoted by a. 



The first of the equations (6) gives 



+ e&quot; mt n sin nx cospy cos qz + -^ cos nx cospy cos qz = 0, 



or + n tan nx + ^=0, 



K 



an equation which must hold when x = a. 



It follows from this that we cannot take any value what 

 ever for n t but that this quantity must satisfy the condition 



nata&amp;gt;una -^a. We must therefore solve the definite equation 

 J\. 



e tan e = -^a, which gives the value of e, and take n = - . Now the 

 J\. & 



equation in e has an infinity of real roots ; hence we can find for 

 n an infinity of different values. We can ascertain in the same 

 manner the values which may be given to p and to q ; they are 

 all represented by the construction which was employed in the 

 preceding problem (Art. 321). Denoting these roots by n^n^n^ &c.; 

 we can then give to v the particular value expressed by the 

 equation 



cos z 



provided we substitute for n one of the roots n v n z , n 3 , &c., and 

 select p and q in the same manner. 



335. We can thus form an infinity of particular values of v, 

 and it evident that the sum of several of these values will also 

 satisfy the differential equation (a), and the definite equations (). 

 In order to give to v the general form which the problem requires, 

 we may unite an indefinite number of terms similar to the term 



cos nx wspy cos qz. 

 The value of v may be expressed by the following equation : 

 v = (a t cos n^x e~ kn & + a 2 cos n z x e~ kn ^ + a 3 cos n 3 x e~ *&quot; & + &c.), 

 (b l cos n^y Q-IM + ^ cos n ^ e -kn?t _j_ 3 CO s n$ e~ kn * H + &c.), 

 (Cj cos n^z e~ kn ^ + c 2 cos n 2 z er*&quot;** + c 8 cos n s y e~ kn H + &c.). 



