440 
MR. A. B. BASSET ON THE EXTENSION AND FLEXURE OF 
These equations will not enable ns to solve the problem in hand; in order to do 
this we require the equations of motion in terms of the displacements, and also the 
values of the sectional stresses in terms of the same quantities. 
6 . The values of the couples, and also the values of M^, can be obtained by 
direct calculation ; but the values of Tj, To cannot be so obtained, since they involve 
the quantities Ah and AJi^, which are unknown, and which cannot be negdected. We 
shall, therefore, be compelled to find the expression for the potential energy, and 
employ the Calculus of Variations. 
The following results will, however, be necessary hereafter. If P', Q', Pd, S', T, 
U', be the stresses at the point a + li, z, (f), we have 
Tpt B(f) = [ P' -fi /d) S(f) dh' 
J — h 
, fdV 
whence 
To = 2/;Q + ^ 
dr 
cPQ 
dd 
3 a \dr 
, 170 , 2 nJd /dm., 
M,= 2«/»3 + J«A3(_ + 
= 2 n/('aT 3 fi- - 3 - nid 
dA. 
dr 
= -|/d 
dQ. 
dr 
F • 
\dr 
- I nld 
/ dm 
dr 
'-’1 + 
dr / ^ a 
^3 
Ho rzz -I nld 
. ( 12 ). 
From the third, fourth, and last of these we see that (Mg — Mj) a = Hg, 
to be the case. 
Let 
^ = 0*1 + E (o-j + o-g), B = 0-3 + E (cr^ -b o-o) 
^ = X + E (X + p,), df? = p- + E (X + p) 
Then, in the terms multiplied by Id, we may put 
P = 2 n.^, Q = 2 nB', 
as ought 
• ( 13 ). 
= 
dQ} 
dr 
= 2udf, 
