CIKCULAli CYLINDERS UNDER CERTAIN PRACTICAL SYSTEMS OF LOAD. 217 
Hence if ajc is very small, v may be large, and tlius although the tirst term in 
rU ^/6 cancels 4 - 5 - yet the terms in curled brackets will become indefinitely large 
compared with the other terms. Thus, for a very long cylinder. 
Avhere Ey is the true Young’s modulus. 
On the other hand, for a very short cylinder, cja is very small, and ^ feeing of the 
order ^ — X — - S .. , the two leadino’ terms in the numerator and denominator 
77^ Of 7 1 ^ 
of the right-hand side of (117) are negligible and 
y ' _— _ 1 _ y 
This is identical with the modulus of compression for a cylinder which is prevented 
from expanding laterally by a constant pressure applied to the sides. So that we see 
that for a very flat disc the eflect on the modulus of compression is tlie same, whetlier 
the lateral expansion be prevented by means of shearing-stress over the flat ends, or 
by hydrostatic pressure over the curved surface. 
The apparent Young’s modulus for intermediate cases will Ije between those two 
values (these, for uniconstant isotropy, being Ey and G/'oEy). Thus, in the given 
example, where y = 2/3, 
Ey'= l-04U8Ey. 
Poisson’s ratio comes out to be apparently ‘2690 instead of '2500. 
Thus the errors in the values of Young’s modulus and Poisson’s ratio, as deduced 
from an exjieriment with cylinders under the given conditions, Avill lie 5 per cent, 
and 7‘6 per cent, respectively. 
§ 2 G. Solution involving Discontinuities at the Ferirneter oj the Plane Ends. 
In § 13 it was stated that a solution, obtained by methods strictly analogous to 
those used in that section, but which neglected tlie condition that the shear rz should 
be continuous at the jierimeter of the plane ends, could be found. 
It seems of interest to give, for purposes of comparison, the expressions for the 
displacements and stresses, deduced from this solution. They are : 
u — -j- ug'^/o -p d" ^ (^'^’) — (At) cos Iz 
k 
IV = wy -p icpA/S + - { 7 ' bj (/‘■'O + y ^’li (A-r) \ sin k: 
k 
2 I' 
^ ( 118 ) 
VOL. GXGVili. —A. 
