126 
PROFESSOR 0. NIVEN ON THE CONDUCTION 
Similarly, when n—m is odd, the values of k are the roots of 
&, = 0,.(24) 
and we have to show that these roots converge to fixed values. 
But before entering on this discussion it will be convenient to take up the con¬ 
sideration of equation (19), and show that it can be satisfied by a definite series of 
known functions corresponding to the same values of k; and, preparatory to doing so, 
I shall digress briefly into the solutions of the equation 
ffl 2 
d/r ' r 
n(n-\- 1)\ 
r 2 ) 
R= 
0 . 
My reason for doing so is that, for the transformations which follow, a connected 
view of the properties of these solutions is necessary. 
7. If we write x—\r, the equation may be written 
dHl 
~d7~ 
+!§+p 
n.n + l 
R=0. 
We shall now show that the equation is satisfied by 
-r, /I d V" sin x ~ (1 d\ n cos« 
B=af U *) or b y E = T '=' r U&, 
• (25) 
If we write "R,=af.w„, we find 
d?u n 
dv? 
+ 2(„-M); 
du„ 
dx 
+ u„ — 0. 
If we differentiate this equation, we may readily put the result into the form 
d* ( 1 du n \ . 1 d (Id \ 1 du n 
-n,- - — +2(»+2) — • - u n +- —=0; 
clxAx dx v x dx\x dx / x dx 
and, comparing this with the former equation, we find 
1 d 
/ M/i+\ — 7 
(26) 
/1 d \ )r 
We thus obtain, generally, u„=[ - ) u 0 
And it is easily seen that u 0 is given by 
