228 Proceedings of the Royal Society of Edinburgh. [Sess. 
for 
(T ^ TJ) = ^[|,x(2Vq„d„)^ - ^DVK] 
CUrn 
= - d„Vq„VqA) - i(Sd„)V(K)] 
v^m 
^-(E + A^q^B), (14) 
where the summations include all values of the suffixes n, s, the differ- 
entiation of terms such as ( — d^Vq^A^q^d^) being performed by means 
of (11), since ( — A^q^^Vq^) is a self -conjugate operator; and that of cross- 
products, such as : — 
(-d^Vq^^A^q^dg) by means of (10), writing a= - A"q^A"qA 
11 C 
A D 
Thus, if in the figure the unit segment is removed from the position 
AD (at which (14) has the value — (E + A^q^B) to tlie parallel position BC 
(at which (14) has the value — (l + ^ry.) (E-f-A^q^B), AB^br), then the 
total increase in L is given by 
8 L = - 8 r V . Sd,^( E + A^q^B ) . 
It will now be convenient to suppose (as we may without loss of 
generality) that the mth set consists of but one tube, so that Sd^ — d^ and 
is in fact a unit vector. 
Then 
- 8 rv.d 4 E + A^q^B), . . ■ (15) 
and in applying the axial differentiator bry we must remember that 
neither d^ nor q^ as they occur explicitly are to be considered variable. 
But to preserve the continuity of the tube we require to introduce the 
segments AB, CD, as shown in the figure, so that, again applying (13), we 
have the change of L due to this cause. 
82 L==d^V.Sr(E-f-Vq^B;, .... (16) 
in which q^ is variable (but not d^). 
Hence 
8 L = -f 62 B = • (E -f Vq^B) - v . d^(E + \ q^^yj , 
