CIRCULAR CYLINDERS UNDER CERTAIN 
PRACTICAL SYSTEMS OF LOAD. 
151 
as I am aAvare, applied his general solution to the problems of tension and compression. 
He does give one example of torsion, which he obtains by applying an arbitrary 
system of cross-radial shears across the flat ends. Such a system, \Ye have seen, 
would not usually correspond to w'hat occurs in practice. 
Mr. Chree ha,s written several other papers (“ On some Compound Vibrating 
Systems,’’ ‘ Camb. Phil. Trans.,’ vol. 15, Part II. ; “ On Longitudinal Vibrations,” 
‘Quarterly Journal of Mathematics,’ 1889; “ Longitudinal Vibrations in Solid and 
Hollow Cylinders,” ‘ Pliil. Mag.,’ 1899 ; “ On Long Piotating Circidar Cylinders,” 
‘Camb. Phil. Soc. Proc.,’ vol. 7, Part VI., &c.), which deal with the solutions of the 
ecpiations of elasticity in cylindrical co-ordinates, with special application to vibra¬ 
tions and rotating shafts; hut 1 cannot find that he has anywhere returned to the 
statical problem and its solution by means of sine and C(.)sine expansions. 
[jJctoher 3, 1901.—Professor Schief (‘Journal de Liouville,’ Serie 3, V(J. 9, 1883) 
has attempted the solution of the prol:)lem of the cylinder compressed Ijetween 
parallel planes, which is one of those treated <jf in the present })aper. His solution 
is expressed in a series, not of circular functions, but of liyperbolic sines and C(.)sines 
of nz, the successive values of n being obtained as roots of a certain transcendental 
equation. This enables him to satisfy the conditions at the curved surface, but the 
arbitrary coefficients are finally determined by the conditions over the plane ends. 
He assumes both the radial shear and the molecidar rotation in a diametral plane 
to be given by known functions, f\v) and F(r), over the plane ends, and from these 
he succeeds in obtainiim’ the coefficients. As he lias onlv a sinule set of the latter 
O O 
left to carry out the identification, his functions f[r) and F(r) are not really inde¬ 
pendent. Theoretically only the shear J\y) should be required, and in a practical 
problem even this is unknown, tlie total pressure being all that is given. Tlie actual 
distribution of this pressure does not appear to enter into Professor Sciiiee’s solution. 
Also tlie fact that the values of n are roots of a transcendental equation singularly 
complicates the solution from a numerical point of view, and Professor Sciuff apjiears 
to have made no attempt to translate his results into numbers.]^ 
It has therefore appeared worth while to apply the solutions involving circular 
functions of 2 to problems such as those sketched above. 
Of each of these I have given a concrete numerical example. Indeed, the greater 
part of the work has been spent on these numerical examples. The labour of calcu¬ 
lation has in most cases been considerable, owing to the slow convergence of many of 
the series involved, wdiich has necessitated special methods of approximation. 
* Since writing the above, I find that the problem of the circular cylinder under a symmetrical strain 
has been considered by J. Tho.mae in two papers (“Uljer eine einfache Aufgabe aus der Theorie der 
Elasticitiit,” ‘ Leipzig Berichte,’ vols. 37-38). The author has used expansions in sines and cosines of U, 
but, as far as I can make out, the only problem he considers is tliat of the vertical pillar under its own 
weight. 
