55 2 
Mr. Airy on the 
rrr ^ 
the common notation / / / 2 • ^ • da . da'. dpt'. The 
limits of ^ are - — 1 and 4- 1 : those of u are 0 and 2 tt : and 
those of a are o, and the value of a at the external surface, 
which we shall call a. 
(4.) To perform these integrations, it is necessary to ex- 
pand the expression for % in a series of powers of ~ or ^ : the 
former of course must be used for the strata interior to the 
point in question, and the latter for the strata exterior to that 
point, that the series may in both cases converge. Suppose 
_ Q 
(O) 
then , — ■ ■ 1 — — — — « 1 
V -|r 8 — 2rR (up 1 -f- y i — v 1 — cos (u 1 — r • 
Q ( ° ‘ ^ + Q (2> * ^ + Q (3) '~4 + &c - for the former case, 
or 
__ Q 
(o) 
R 
+ Q ( ° • 4 + Q (2) - 4- + Q (s) - & + &c. for 
the latter, (Q°\ Q ( '\ Q^,&c. being the same in both series): 
(z) . .... 
then Q satisfies the following equation, (liv. 111. n°. 9 ) 
0 ““ d« I 1 t 
dQ 
| ~i~ - — — - 
f* ‘ T 1 
d'Q 
+ i.i+ 1 Q 
S) 
which is likewise true if we put p for p and u for a. And 
the value of V now depends on the integral 
( since p is a function of a alone ) , for the interior strata, i 
being made successively =0, 1, 2, 3, &c. ; and on the inte- 
gral r 
made 
9 
(0 
/ R 1 1 
O, 1, 2, 3 , &C. 
d R 
da 
, for the exterior strata, i being 
J + 3 
( 5. ) Suppose now R reduced into a series of the form 
