2G0 PEOFESSOB G. H. DAEWIN OX THE STABILITY OF THE PEAE-SHAPED 
The surface r = constant is an ellipsoid similar to J with squares of semi-axes 
reduced in the proportion 1 — 2 t to unity. Therefore the volume enclosed between 
the two ellipsoids is 
I/. 
[3. 
rr" — 
But taking the limits of 6 and (f) as to 0, so that we integrate through one 
octant and multiply the result by 8, we have another exjjression for the same thing, 
namely, 
jdu = ^ j j [cPr |nr3] cW c^. 
Therefore equating coefficients of powers of r in the twm expressions. 
in 
in 
The first of these will be of use hereafter, and all three afford forniula3 of verification 
the numerical work. 
§ D ctermlnation of h ; Definition of Symbols for Integrals. 
Tlie pear being defined by r = — e>S’g — '^.ffSf, with all the fin of order 
1 
excejiting fi which is zero, we have at the surface of the pear to the fourth order 
+ 2Sfif^S,Sy + {tffSfY, 
r^=-e^ {S,y - {S,YSf, 
r-i _ ^4 
In all the integrations which follow, and especially in the present instance in the 
determination of the volume of the region H, it is important to note that (h, H are 
even functions of the angular co-ordinates, and that therefore the integral of any odd 
function of those co-ordinates multiplied by any of these functions will vanish. 
When the odd functions are omitted we may integrate throughout the octant defined 
by the limits Ttt to 0 for 0 and (^, and multiply the result by 8. 
Then, only retaining terms as far as e®, we may in finding the volume R take 
T = — ^f\Sg, i only even, 
t" = e- {S.y‘'- -h '2SefyS.^Sf, i only odd. 
t" = 0 . 
To the cubes of small quantities we have, therefore. 
