472 
MR. A. B. BASSET ON THE EXTENSION AND FLEXURE OF 
If in the fii’st surface integral in this equation we substitute the approximate values 
of the coefficients of d 8 K/«^ &c. from (23), which we may do since this integral is 
multiplied by Id, and then substitute the values of SWg", SC, 811, and SU in ( 10 ), it 
will be found that all the terms involving f/SK/^/d, cZSK/cZ(/), and 8 X + Sp, cut out; 
we are therefore no longer concerned with them, and the value of SWg" reduces to 
the last line ; on this understanding we may write 
8 Wg'' + 8 W;' = t'f II (^ 8 X + B Sp + ITI7 Sp) f/S ... (28). 
The variation of the right-hand side of (28) might at once be written down from 
(27) by substituting and for 5c, and p ; but it will be more convenient 
to present the I'esults in another form. Taking the first term, and integrating the 
second differential coefficients once by parts, we obtain 
(1810 
U 
sin 6 
d(f) + I 
d d 8iv 
(^ sm 6) 
d6 
dd 
sin 0 Siv 
sin 0 Sk| d6 dj>. 
Treating the other terms in a similar way, and adding to the result from ( 21 ) that 
portion of SW^ which depends upon Id, and finally replacing the coefficients of 
dStvjdd — Su, &c., by their approximate values from (23), the final result will be 
- I(tJ - ^“) + i {le - Sf)} «sin edj, 
4:nJd 
1 d Bid 
\sin 6 dcj) 
dSio 
8 y) + 5 nr (— Su ] j- add 
Bv 
1 d Bio 
sin 6 d^ 
+ E« {iv —■ Z) 8 K 1- dS (29). 
This result enables us to test the accuracy of a poi'tion of our work, and the funda¬ 
mental hypothesis on which the theory is based; for if we substitute in ( 10 ) the 
expression (29), and also the value of 8 |i from (19), it will be seen that we have 
reproduced the values of the couples which are given by ( 7 ) ; also comparing with 811, 
the line integral parts of 8 W 3 , given by (27), the line integral parts of 8 Wg''and 8 W/, 
which, as we have explained above, are obtained from ( 21 ) by changing certain lettei’s, 
we see that we have also reproduced the values of M], M 3 , given by the third of ( 6 ). 
We may, therefore, omit the couple terms, and also the terms in M; also, since we 
