356 
Proceedings of the Boyal Society of Edinburgh. [Sess. 
(6) In accordance with the assumptions laid down, the last two equations 
must be exactly the same. 
Expressing this condition, we have 
3[r 2 cos 2 00 2 cj> 2 ] — [r 2 sin 2 #<f> 4 ] 
= [r 2 sin 2 0 cos 2 (9 </> 4 } - [(cos 2 0 + 2 )r 2 # 2 <f> 2 ] - [r 2 # 4 ] 
In order to evaluate these mean values, let us put 
( 6 ). 
Then 
Now 
q cos i(/ = rO 
q sin t f/ — r sin 6<p 
[r 2 # 2 </> 2 sin 2 <9] — 
9. 
t- sin 2 1 1/ cos^ if/ 
if* £ 
sin 2 2i b 
4 r 2 T 
2n 
[sin 2 2ifA = — / sin 2 2i J/difr = \ . 
2 7TJ 
(7). 
Likewise 
and 
[?‘ 2 6 ,2 </> 2 sin 2 ff\ = -g- 
r 
•v*2 
[r 2 sin 4 #</> 4 ] = | 
t’ -2 ^ 4 ] = f 
T 
2tt 
for these depend merely on the evaluation of j sin 4 \Jsd\/s and j cos 4 \fsd\js. 
By differentiating the equations (7), squaring and adding, and using 
the relation (6), we can obtain also the relations 
and 
\r 2 0 2 (f>' 2 ] = T y 
[r 2 sin 2 04 > 
41 _ 5 _ 
1 6 
v 
\ 
r 2 
_r 2 _ 
, 
(8) 
but these relations will not be required. 
Yd 2 
(7) Expressions such as — (rr&) 
-til 
and 
L ^V 2 sin 
when 
evaluated are found to all vanish identically in accordance with the 
hypotheses. Now let ds 2 — r 2 d0 2 -\-r 2 sin 2 0d(p 2 , so that ds is a small 
element of length perpendicular to the straight line AB ; and let us add 
together both sides of equations (4) and (5). 
