■2o(j 
PKOFESSOK G. II. DAIGVIN 
ON THE STABILITY OF THE 
PEAB-SHAPED 
abridged notation to specify them. I may denote the lost energy of J, inclusive of 
rotation, by \ JJ\ the mutual lost energy of J and of the region it, considered as 
filled with j^ositive density, by JR ; the lost energy of the region R by \RR. 
The moment of inertia of the pear is A, and it is equal to Aj — A^, the moment of 
inertia of J less that of R. 
Then 
R = IJJ -JR A- ^RR + i (Aj - A,) - 1 (0)3 + So)2) MJ, 
where 
JR = [ [F; + |o)3(y2 _|_ 'Ay\pdv, 
J r 
A = f (r + pdv, A, = [ ( f + z^) pdv, 
•J •' >■ 
and 
ARR = f V,pdv. 
J r 
If ^DD denotes the lost energy of the double system described above, v'e clearly 
have 
ARR = ^{C - R){C - R) + CR - ^CC = ^DD + CR - ^CC. 
We require to evaluate E to the fourth order; now d is at least of the second 
order and dd of the fourth order; hence d^ . hod is at least of the fifth order and 
negligible. 
Hence, finally, to the required degree of approximation 
E = ^JJ - JR A-CR- \CC + IDD + ^ {A, - A,) - AMdW . . (3). 
It will appear beloAv that d is not even of the second order, so that the last term 
will, in fact, entirely disappear, although we cannot see at tlie present stage that this 
will be so. 
§ 3. Expression for the Element of Volume. 
The parameter /3 of “ Harmonics ” is connected av 
by the equations 
1 — _ 2 p _1 — k'^ 2(3 _ k'" 2(3 _ 
1 +(3 “ 1 + “ iTf ’ 1- J ~ 1^/3 “ 
There will, I think, be no confusion if I also use /3 in a second sense, definiug it by 
the equations 
ith K of the “ Pear-shaped Figure 
sin yd = K sin y, 
0 r\ I v) • 0 
cos- (3 = I — K' sm~ y. 
