18 
ME. J. H. JEANS ON THE INSTABILITY OF THE PEAR-SHAPED 
These equations are not, as might at first be thought, a system of three simple 
equations determining p, q, t\ They will be found to be of the type known as 
“ porismatic that is to say, equations which are inconsistent unless the coefficients 
satisfy a certain relation, and such that, when this relation is satisfied, the equations 
have an infinite number of solutions. 
Let us, for brevity, write the equations in the form :— 
4p + 74 q + k'\ X = % 
4p + k s q + k 2 t — 3& 2 
Arp + 7d,q + k r $ = 3^3 
I 
(80) 
Let us use also the abbreviated notation of the previous paper (p. 50), such that 
f* \d\ _ f* A d\ _ f Xd\ 
Cl Jo A ABC’ ° 2 " Jo A A 2 C’ Cs ~ Jo AA 2 B’ 
Then, by simple transformations of the integrals, we obtain 
1 ! 
V — 
lb , — 
4 = 
1 26 
4a 2 \a 4 
-c 2 - 
-C 8 b 
k\ 
1 
= 4 b 2 
I'; = 
3 
k'o 
1 
120 
— 3 Co 
4 a 3 
4 b 2 
\a 2 6 2 
h = 
3 
= q 2 C.,, 
4 a 
k'g 
1 
“ 4 b 2 
Ci, 
l J ' ~ 
3 — 
1 
, 2 C 2’ 
4 c 
4c 2 
■ ( 81 ) 
1 , 20 
A 9 \ 9 2 
4 C- \Oj G 
With these values of the coefficients, it will be found that equations (7l)-(73) of 
the previous paper reduce to 
2Aqa + 2k\j3 + 2 k'\y = 04 
2 La + 2/d 2/ 3 + 2 k" 2 y = 0 l,.(82) 
24 a + 2k' Sl 6 + 2k",y = 0 J 
so that 
p = 2a, q = 2/3, r = 2 y ,.(83) 
is a solution of equations (80), when It] = = W = 0. 
W ithout this detailed inspection of the equations, it could have been foreseen that 
this would necessarily be the case. For our general solution 
<j> = eP + e 2 Q + e 3 K,.(84) 
* See ‘ Hobson’s Plane Trigonometry,’ §73, or Wolstenholme, 1 Proc. London Math. Soc.,’ vol, 4. 
