150 
.MR. L. N. G. FILON ON THE ELASTIC EQUILIBRIUM OF 
problem for tension, and corresponds to the case of a bar “gripped” as explained 
above, and twisted. This again is the method by which torsion is practically 
produced in most cases—almost always in laljoratory ex})eriments. 
^ 2. Method of Solution Adopted. Historical Referey,.ces. 
Tlie method adopted has been to olRain symmetrical solutions of the equations ot 
elasticity in cylindrical co-ordinates and to express the typical term in the form 
cos 
sin 
\hz] Xf{r) 
r, (f), z being the nsnal c}dindrical co-ordinates. 
The expressions for the strains and stre.sses, oyer any coaxial cylinder, are therefore 
series of sines and cosines of multi})les of r. The arbitrary constants of the coefficients 
are determined by comparison witli the coefficients of the Foueier’s series which 
express the applied stresses at the external honndary. 
This metliod is not a new one. It has been indicated by Lame and Claeeyeox 
(“ Memoire sm- re([nilihre interienr des corps solides homogenes,’ ‘ C'relle’s Journal,’ 
vol. 7), hut it has been lor the first time worked out with any completeness Ipv 
Professor L. Pochhammer (“ Beitrag zur Theorie der Biegung des Kreiscylinders,” 
‘ C'relle’s Journal,’ vol. 81, 1876). Professor Pochhammer obtains the general 
solution of the elastic equations for an infinite circular cylinder sulqect to any system 
ol surface loading, repeated at regular intei'yals. This he a})plies to the case of a 
built-in beam. The solution is not restricted to he .symmetrical al)out tlie axis of the 
%j 
cylinder, Imt is perfectly general. The complete accurate expressions are, hoM’ever, 
(juite ninvieldy ; but, as the residt of expanding the functions inyolved to the first 
t^yo or tliree teiins, Professor Pocihhammer oljalns far more manageahle expressic-tiis, 
which he is eyentually able to Identify \vitli tlmse previou.sly given V)y Nayier and 
DE Saint Venant for more .special cases of loading. It is to be noted, howeyer, that 
Pochhammer restricts himself .solely to the case of liendlng, and that his approxima¬ 
tions depend upon the ratio of diameter to lengtli being a small quantity. 
’fhe same general expressions have been independently arrived at by Mr. C'. Chree 
(“The Equations of an Isotropic Elastic Solid in Polar and Cylindrical C'o-ordinates, 
their Solution and A])pllcation,” ‘ Camb. Phil. Trans.,’ vol. 14). Here, again, the 
solutions are not restricted to 1)e symmetrical. The .symmetrical terms, however, 
agree Muth the solutions of the present papei-, but the latter are obtained by a process 
slightly dirterent from that of Mr. Chree. Mr. Chree has also given a solution ol 
the .symmetrical case })roceeding in poMmrs of r and .t. Using each form of solution 
independently, it is not po.sslffie to satisfy the condition tliat there shall be im stress 
at all on the curved surface; this is effected in the second prolJem of this paper, hy 
means of a comhination of the two ty})es of solution. 
In the paper referred to, Mr. Chree, like Professor Puchha;mmer, has not, so far 
