268 PROFESSOR G. H. DARWIX OX THE STABILITY OF THE PEAR-SHAPED 
8. The Terra — ^Md-or. 
In the Jacobian elli 2 :)Soid 
oJjj “ 
In the present notation this is 
7,Sa,= /I + cos= /3\ ^ I ^ 2 
Hence 
iLI/ \ sin- yS / /i 
- = - 1 
J/ 
3 sin- yS 
_ _ '-' / g- _ 3 g \ 
1 + 
I now make the following definition 
,Si := sin ^ (1 — K ~ COS' (f))\ 
z = kS,. 
so that 
Then 
Md = zp dv = \ zp dv — zp dv = — \ zp dv 
}j—r Jj Jr Jr 
I |[d) - S^dr de d(f) 
= M (~Jk (I [4. {cS, + sy?s,‘) + (S,n S, cW # 
/k \3 
Tlierefore to the required order 
- ^MdW = - 
31/2 gppy3 
.. 
( 20 ). 
We ago,in note that this term in the energy does not introduce any term with a 
coefiicient e~fi. Hence thus far the whole energy for harmonic deformations of odd 
order is of the form Le"** + M {fdY. 
§ 9. Double Layers. 
It remains to determine the value of ^DD in the energy, and for this purpose we 
must consider double layers, according to the ingenious method devised by 
M. POINCAEE. 
Let a closed surface S be intersected at every point by a member of a I'amily of 
