n28 
PP.OFKSSOR a. H. DAPAVIN OX THE PEAP-8ITAPED FTGTT.E 
§ 8. The 2 :)€aT-sha'ped Figure of Equilibrium. 
By (21) the normal disjilacement hn for the third zonal harmonic deformation may 
he written 
hn — e 
- (3 - MFir - dq-<r siid 7 ] 
- rsin^Y) (.-,’2 /cos^ 7 + ?/- 'Ah+ r)’ ’ 
.subject to the condition 
V- 
+ .V 1 O ^ 
^ 
ITie expre.ssion ha.s l>een arranged .so tliat when :c = ?/ = 0, ’ =1 c. we have hn = e. 
Hence + e and — e are the normal displacements at the .stalk and blunt end of the 
]*)ear respectively. 
Tn the section y = 0, this may he written 
hi — 
c cos 7 (*d — Crq~) 
/-■> • 0/0 0^*0 \h * 
The nodal points are given by - = i 9 = i ‘75805^. 
Tn the section x — 0, since F = cr\ -^ 
— 4^- 
hi = c 
A(4-%2) 
o o 
CFK'^ 
it may be written 
/ o o o o\ 
•t {k~z~ — c-q~) 
(c- — /c-.r Bin- 7 )- 
The nodal ]’)oints are given Ipv - = 4 “ = 4 ‘788980. 
C‘ K 
The section .'j = 0 is obviously another nodal line for all sections. 
By means of these fornudfe it is easy to compute the normal displacements from the 
surface of the critical Jacobian. 
Tlie figure opposite showing the three sections x — 0, y = 0, ^ = 0, is drawn 
from these formulm, the dotted line heino’ the critical Jacobian and the firm line the 
pear. The scale of the normal displacements is, of co\irse, arbitrary. 
Comparison with M. PoixoAEi'fs sketch shows that the figure is considerably longer 
than he supposed. 
In this first approximation the positions of the nodal lines are Independent of the 
magnitude of c, and they lie so near the ends that it is impossible to construct an 
exaggerated figure, for if we do so the blunt end acquires a dimple, which is ab.surd. 
It might have been hoped that .such an exaggeration would afford us some idea of the 
mode of development of the pear. 
