148 
PROFESSOR C. NIVEN ON THE CONDUCTION 
we obtain 
rp p » ,/p » rn~ 
< x • -«(»+ 1 )P-‘. 
X 2 c 2 f + V/P> 2 <A;=(&, — n(n+l)) ( + Vp^ 
• -l - -l 
= ft, ■—n(n +1)) (— 1 ) r a/j m r , n=m+2r, 
in which p denotes any even integer. 
But 
A/=ct/'P „r —«/P m ' n+ ~+ 
hence, remembering equation (63), 
X 2 fi 3 j i/3-P$r'dv= —[TO(m+l)a 0 u' 0 y ra °+a 1 a , 1 (m+2)(m+3)y/ z+2 + . 
= — {p>p'}> sa y- 
] • (67) 
The expansion of zA9>/ therefore becomes 
W= — 
e 
q o .Milo i, 
_ (0, of* ^ (1, 1) * 
• ( 68 ) 
18. These investigations and developments place us in a position to deal with the 
boundary condition due to radiation; we may always put 
*/l- eV^'=^°+^ 1 +y/A 3 -f.(69) 
and all the g ’s may be expanded in ascending powers of <? 2 , beginning with e~, except 
g n n , which commences with e°. The general equation of conduction is to be satisfied 
in this case by the series 
V=(A cos sin m<f>)e- xt «(C^°n 0 +C 1 9- 1 n l + C^*n 2 + ■■-),- • (70) 
in which the same X occurs throughout. 
At the boundary, where £=Xb, 
^ 0 {~di^~ S,0 ° n ^) G^° lni + + • • • = 0, 
Co^i°B u +C 1 ^^-l-qi 1 ni^H-C,^ 1 ~no+ .. . =0, 
C 0 ,^° n o+Ci^htq+ +9* n i) + • • ■ =°> &c - 
