CYLINDRICAL AND SPHERICAL THIN ELASTIC SHELLS. 
477 
portion of a sphere bounded by two meridians and two parallels of latitude, and 
whose edges are free, the boundary conditions along a parallel of latitude are obtained 
by equating the right hand sides of the first and fourth of (33) and of (42) to zero ; 
whilst the boundary conditions along a meridian are similarly obtained by equating 
the right hand sides of the first and fourth of (34) and of (43) to zero. 
21. If the shell is supposed to vibrate in such a manner, that its middle surface 
does not experience any extension or contraction throughout the motion, the equations 
of motion can be obtained by taking the variation subject to the conditions of 
inextensibility, and introducing indeterminate multipliers. 
22. It will now be convenient to make a short digression for the purpose of con¬ 
sidering some of the quantities involved. 
Let P be any point on the deformed middle surface whose undisplaced coordinates 
are (a, 0, ^). The coordinates of P after deformation are 
R,= a-b^^, B = 6 uja, ^ = (f) vja sin 0 . . . (44), 
and since u, v, w are functions of d and </>, the elimination of the latter quantities from 
(44) will give a relation between P, @, $, which is the equation of the deformed 
middle surface. 
If Pi be the radius of curvature at any point of a meridian section after deformation, 
and P the perpendicular from the centre on to the tangent at that point to the 
deformed section, 
Now 
I 
Pi 
1 ^ 
R 
P2 
I 
R2 
1 + f—Y 
^ \RdBj 
If (chvld9f 1 
R3[ ' R^l + dujadd)-]’ 
and therefore, neglecting cubes of displacements, 
_ I / aia\^ 
Also 
dR = do, 
ecu 
whence 
II I (d^w \ 
a ^ ~ ^7 • • • 
.... (45), 
which gives the change of curvature along a meridian. 
