48 
MR. HOPKINS’S RESEARCHES IN PHYSICAL GEOLOGY. 
which gives Z n — 0, except for n = 0, or n — 2 ; aiid 
cos 2 e — 4 ) 
Zo = ^T’ Z 2 = C£,(< 
also 
Hence 
, r , f 1 /N 0 N 2 r 2 \ , N 2 r 2 9a } - a ~t 
U ~ C g l { 3 \(N 0 ) (N 2 ) ' a*) + (N a ) * a* C0S ~ 6 j 2 ** * 
It remains to determine N 0 , (N 0 ), N 2 , and (N 2 ). We have 
N re = f Q COS 7 T COS Co') sin 2w+ 1 a dco ; 
or putting cos co — x, 
N m = x f^ x COS 7T . x) ( I — X «)* (1 X ; 
and performing the integrations for n — 0 and w = 2, we obtain 
2 a 
N 0 = — *— sm-T, 
, T /48 «, 5 16 «, 3 \ . r 48 a, 4 r 
N 9 = ( —5 • -V — -3 • Hr J sin — ■r 5 • — cos — t 
> y^b r b ^6 yii ) a Jr 4 r 4 Gj 
and putting r — « l5 we have 
(N„) = 2, 
(N 2 ) = ~ 
Hence 
Nl 
(N 0 ) 
N, 
(N 2 ) 
1 
a x . 
— sin 
r 
— • 
7 r 
r 
“1 ’ 
/1 
< 
3 r 3 / 
r 
In 
1 sin — 
a, 
T — 
—4 COS — T ; 
r 4 a, ’ 
and by substitution, we have the complete value of u ' ; and for that of u we have 
fa, . r , e,/N 0 N 2 ? - 2 \ N 2 r 2 
“ C { 7rr Sin cj T + 3 \(N 0 ) ~ (N a ) a, 2 / + £ i (N 2 ) «, 2 C0S ' ^ j 
or putting 
«i 
C 
= G, 
we have for the equation to the isothermal surface, of which the temperature is u at 
the time t, 
■1 / N 0 N 2 r 2 \ 
, 2 X _ Nj r 2 
