10 
MR. J. H. JEANS ON THE INSTABILITY OF THE PEAR-SHAPED 
In this solution, fi> already contains a term involving A, namely that given by 
expression (43). Combining this with the remaining term in expression (45), we find 
that the new solution can be put in the form 
fit i + -g-7 -pcibce s 
0 ( f<f)X 
ax' 1 AA 
A dX 
(46) 
where fi> 0 is the old solution fi> with the terms in A omitted. The last term in expres¬ 
sion (45), being proportional to e*<p, is of the fourth order of small quantities. Thus 
in a solution as far as e 3 only, this term may be omitted, and fi> = fi> 0 will be a solution. 
In other words the term Z may be omitted entirely from equation (42), and the 
remaining terms will still give an accurate solution for v 3 . 
Omitting this Z-term, we obtain for the third order terms, 
«j+,A’s = U 3 +/(w,+/w',) 
= AD s UP*)-iD(PQ)4-E 
- i/{*D s (■ F) - JD 3 (PQ) + DR} 
+ */* -*ir(PQ) + iD s B}.(V) 
This completes the solution of the general potential problem. 
* 
« 
Potential of the Pear-shaped Figure. 
9. Collecting the results obtained in §§ 3-8, we have found that as far as terms of 
the third order of small quantities, a value of <j> which satisfies the necessary 
differential equation (6) is 
4, = k (u, +/v,) + /r (u, +,/;■») + e~ ( m , +fo 3 ) 
= e[P-*/DP] 
+^[Q-JDP*+/{*D*(P^-*DQ}+/»{-iA»D , P*+AD , Q}] 
+ « 3 [R-J/DPQ + T feD 2 F-i/ { S },D*P-i-DVQ + DR} 
+ A/ 3 {AorD‘P a -*D s PQ + iD ! R}].(48) 
At the boundary A = 0, this value of 0 reduces to 
0= eP + e 2 Q + e 3 R,.(49) 
and since P, Q, Pt are entirely at our disposal, this is capable of representing the most 
general displacement possible, as far as the third order of small quantities. We have, 
however, to save the printing of additional terms, already assumed that P is of a 
degree not higher than the third in x, y, z and Q of a degree not higher than the 
fourth. Subject to these limitations, the potential of the ellipsoid deformed in any 
way can be obtained by inserting the value (48) for <p into equations (3) and (4). 
