CYLINDRICAL AND SPHERICAL THIN ELASTIC SHELLS. 
461 
Since the numerator of this fraction is an even function of (f), it does not change 
sign with (f ); also the numerator is always positive between the limits ^ tt and — ^ tt, 
and its maximum value occurs when </> = 0 and is equal to ^ tt — 1, and its minimum 
value occurs when xf) = 177 and is equal to zero. We, therefore, see that when xf) = 0, 
R _ E 3a 
(7„ a ’ 
and when </> = l tt, 
R 
Vo 
E 
a 
Since the thickness of the shell is supposed to be small compared with its radius, it 
follows that the change of curvature is large compared with the extension of the 
middle surface, except when a (^tt — <^) is comparable with h, i.e., in the neighbour¬ 
hood of the straight edges of the shell; and therefore at all points of the shell whose 
distances from the edges are large in comparison with its thickness, the terms 
depending upon the product of the change of curvature and the cube of the thickness, 
i.e., the terms upon which the bending depends, are of the same order as the terms 
depending upon the product of the extension of the middle surface and the thickness; 
but at points whose distances from the edge are comparable with the thickness of the 
shell, the extension of the middle surface is of the same order as the change of 
curvature, and therefore the terms depending upon the product of the change of 
curvature and the cube of the thickness are small in comparison with the terms 
depending upon the product of the extension and the thickness. 
We shall now calculate the stresses which must be applied to the circular edges in 
order to maintain this particular kind of strain. From (43) we have 
Gg = f nJi^ E (/X, cTg/a), 
Substituting the values of /x and 0-3 from (62), (63), and (64), we see that the terms 
in (Tg may be omitted, and we obtain 
Go = 
EWa 
1 + E 
— (f) sin (f) — cos xf)) 
(G8), 
which shows that G 2 is positive. 
Also 
Tt = Anh E cTo 
2nJt? 
3 a 
E (2 -f 3E) [x 
EW 
1 + E 
xf) sin xj) .(69)j 
which shov/s that is positive. 
Comparing ( 68 ) and (69) with (61) and (62) we see that ratios of the tension 
Tj and the couple Gg, to Tg and G^ are numerically equal to E/(l -j- E) ; we further 
