146 
PROFESSOR C. NIVEN ON THE CONDUCTION 
17. When the ellipsoid cools by radiation, the equation to be fulfilled at the 
boundary is 
dV 
du 
+ f)C\/ cosh~ a — cos 3 /3V=0, when a = a 0 . 
or 
dV , 1) 
+7 a/ 1— cr cos 3 /3V=0.(59) 
observing that —r—=-=<?. 
cosh cc 0 a 
In its present shape, the process of satisfying- this condition is complicated. If, 
however, we neglect e 3 , the condition then becomes 
The appropriate form of solution in this case is 
V= (A cos B sin m<ft) 
X being given by 
rJQ n h 
~ +r’n«-=0, when £=&. 
ttg A, 
(60) 
If, as before, we neglect e 3 , this equation becomes simply 
dS /t f) ,, 
rff+V“ =0 > 
the same equation as found for the case of a sphere. It is, therefore, only when we do 
not neglect <?A 3 in the expression \/l — e 2 v~, that we obtain results belonging specially 
to the ellipsoid. We must accordingly indicate how the problem in its more general 
form is to be dealt with. To do so we require various general properties of the 
S -functions, to the discussion of which we now proceed. It should be understood 
that these properties are of a purely mathematical character, and have nothing to do 
with the special series of values which the physical conditions of the problem may 
ascribe to X. We have seen that for given values of X and m, the different values of h 
are perfectly definite, and it may facilitate the apprehension of these values to bear in 
mind the approximations which have been given for them in powers of e. 
The following theorems are well known:— 
+ 1 P/PA^=0, [ +1 (P/) 2 cA=P 
(n—m)l (n + m)l __ . n 
{1.3 . . . (2n-1 )}2“> ’ 
say 
• ( 61 ) 
