120 PROFESSOR C. NIVEN ON THE CONDUCTION 
and the boundary condition 
v =° or ^+^=°.(6) 
When the appropriate functions v have been found, in general triply infinite in 
number, to satisfy (5) and (6), the constants A may be determined from condition (3). 
Now it is obvious from the nature of the case that any solution of the equations which 
satisfies (1) and (2) and reproduces (3) when t = 0, will be the solution sought, and the 
same conclusion can be readily demonstrated by analysis. With regard to equation 
(6), it serves two purposes : first in enabling us to select the appropriate form of v, and 
secondly in furnishing the values of X, which determine the types of heat-movement 
which take place. Poisson has shown, in a very elegant way, that the values of X are 
always real, and as his results are of importance as showing also how the constants A 
are to be found, I shall here reproduce them. 
Let v and v' be two functions of x y z satisfying the equations (5) and (6), and let 
V s stand for —+ j then 
clx“ ay- dz* 
(X'~—X 3 ) [ vv'dxdydz = j (v v ~v—v v 2 v')dxdydz 
=1 
, dv dv ', , 
■“s-ts )*■ 
ds being an element of the surface; hence, if either form of (6) be true, 
It follows from this equation that the equation in X cannot have imaginary roots; 
for, v being always a function of X, if X=_p + q v / —1, there will be another root 
\'=p— q\/— 1, and the corresponding values of v will be respectively L + M^/—1 and 
L — My/— 1. Equation (7) now becomes 
j" (L 2 + M. 2 )dxdydz = 0 , 
which is clearly impossible. We may also employ (7) to find A, for 
V (J =SAv 
A=®gf, dE=dxdydz .(8) 
in which the integration is extended throughout the whole of the solid. 
