450 
MR. A. B. BASSET ON" THE EXTEN'SION' ARD FLEXURE OF 
3 [ [I , 13 dhto \ , 
“ T/+ ; j 
= f pM (y 8 m + a hiu) a d(f) + -| ph^ (y Sv + /3 Sw) dz 
~ jf{5 + : S St’ + (^-^ + ^-^ - -) Si(;k^S 
1 dj 
a d(p 
da 
~dz 
1 dfB 7 
a d(f) a 
■ («)■ 
Substituting the values of SW^, SWg, 8Wg', SW/, and the right hand side of (42) 
in (40), and picking out the line integral terras, we obtain the following equations 
for the sectional stresses, viz., 
T; = 
oco 
Mg = 2nhT^ -b ij nldp' 
N, 
Go 
H, 
+ i «VB' + ^ ic + ^ E (aEii - w + Z) 
, 2/ik^ 
3 ^ dz + 2« + 3« dz + 3 p/t 
I nld 
-f 
- I [p + ^ 
(43) 
which give the values of the sectional stresses across a circular section ; and 
— 2nhm + 3 nh^'p 
To 
4.nhU _ Ei? + I nh^^' + ^ E (aEK - w + Z) 
O Ct o Ct 
N,- 
Gi = 
Ho = 
I nk^ (i f + 1 fl) + 2 ^ _ 2 -^ + Y 
\a d(f) ^ dzj da d.z da \d(j) 
- i 
I idd'P 
(44) 
w'hich give the values of the sectional stresses across a meridian. 
If we compare these equations with the third and fourth of (12), with (14), and with 
the fourth and fifth of (H), we see that we have reproduced (i.) the values of Mj, Mg 
given by (12); (ii.) the values of the couples given by (14); (hi.) the values of the normal 
shearing stresses which are obtained from the fourth and fifth of (H), by substituting 
the values of the couples from (14). We have thus subjected our fundamental 
hypothesis to a fairly searching test. It is, however, in our power to subject it to a 
still further test; for if we equate the coefficients of S’ 7 , Sv, Sw in the surface integrals 
in (40) and (42), we shall obtain the equations of motion in terms of the displacements, 
