f 
278 PROFESSOE G. H. DxVPaTIX ON THE STABILITY OF THE PEAR-SHAPED 
This result may also be obtained as follows :—To the first order we may concentrate 
the negative density in the region R on a surface bisecting that region. We may 
then consider the positive concentration C on J, and the negative concentration on 
the bisecting surface as an infinitesimal double layer of thickness |e. We have seen 
that the surface density + (7 is — ppeS.^^ and that e = — peS^ (in both cases to the 
first order only). Thus the density 8 of + (7 is pe, and the thickness r of our layer 
is -^e; the product therefore t 8 is 
CO 
Hence tS = \pe^ = bp S and thus we arrive at the same result as before. 
0 
I now introduce an abridged notation analogous to that used previously, and write 
(Ivq fb'o 
We then have by (26) on the last page 
V ” 27r»,,p cos B cos 7 , _ ^ 
hi 7 dn/3 . 
where e" = N /p'tS'f. 
dV 
ch 
By (22) to the first order 
Assume then 
o _ -■) .T /Cl x -"* _ o / o COS" B COS" 7 
= nv ('%)- = e-t- 
COS- /3 COS" 7 (S.p~ ^ ” 
sin/3 
Multiplying by d>/S7 and integrating, we have 
Hence 
^2 7.3 CO ^ s /’ 2/.2 n * 
€' = n therefore hf = . 
sin Bo<Pf sin B 4>f 
Substituting in (26) 
dii 0 
o cos B cos 7 p/ „ . 
M 
Now 
= - i As ^ TA Q/AA 
0 Yi 
e- = AAA V ^ N/, and 
sin B 0 4>i 
Pi 
dn 
/ ? K c 
N -A Q LS'/ ( N ^ Si 
\ 0 ^7® / V 0 
I 
