290 Mr G. H. Bryan, Application of the Energy Test [Oct. 29, 
Whatever be the nature of the displacement 87/a may always 
be expanded in the series 
ér/a= =u {A cosn@ +B, sin n6}, 
and from equation (2) the corresponding value of 6 is 
66 = > {—A,/n.sinn@+ B/n. cos nO}. 
The total increase in circumference is 
=|" ead@ 
0 
w [27 (1 /dér\? or ddé 
ae sig (ess Ms eg 
al id tf a 7} 08 
= rad {(n®— 2)(A," + B,’)}. 
Hence the work done by the thrust 7’ (per unit length of the 
cylinder) is 
=pmal' {(n’—2)(A,? + B,), 
so that the increase of potential energy due to this cause is 
OW, =—4traT>d {(n? —2)(A,?+B,”)}. 
The volume of the deformed cylinder per unit length is 
a 5 | rag ms 5| (a + 8r)?d (0 +80) 
1p sf Bry, dO 
=f, @ (+5) + Gy) @ 
oe (Giet Cee MORE) eo Ne ole Glole) 
= ae O eent — — 9) se ee 
a {ee tag (G+ a 10} 
(neglecting small quantities of the thard order) 
=a’ — haa" (A,” + B,”). 
sd 
Thus the increase of volume is 
—tra’> (A, + B,”), 
and the wncrease of potential energy due to work done against 
the external pressure is 
SW,=— }ra’P> (A,7+ B,”). 
Lastly, the increase of potential energy due to bending has been 
shown by Lord Rayleigh to be 
6V= 7 ee XY (nv? — 1) (A,? + B,”). 
a 
