77 4 
MR. A. McAULAY ON THE MATHEMATICAL 
m s = ds Ids = Td’Z'/TdX = mT^' 1 i . 
{</>/} = — m ~ l ix^sX — i [XrV^S ( ) xU^ + xU^S ( )"xrV/J] 1 
<£/— [{<£/}]«+& 1 
Then it is easy to show, after the manner of § 39, that 
Bl s = m, [SSp/ {<£/} V/] ffi+ 6 — SSd [ d VZJ (6 + b . . . . 
(4). 
(5). 
( 6 ). 
To see what modifications must be made in equation (9), § 45, consider first the 
first term on the right of equation (6). This contributes to SL for a finite portion of 
space JJSSp/ {<£/} V/cfo'. Put in this V/ = — i'Si'Vj' — f'WV/. Thus 
jjss P '{<!>;} v.'ds' = - [fsf'v/sap/ {$;] i'ds - (fssp/ {<£/} (i'wv/) ds. 
109. We will anticipate somewhat here, as the effect is a considerable simplification. 
The first of the terms on the right involves the vector —Sf'V'.Sp', and this is the 
only term involving space derivatives of Sp' that cannot be transformed into terms 
involving Sp' only. Now the vector coefficient of this vector, like that of Sp' in 
SL + SQSgt must be zero. For — Sf'V'.Sp' is the normal space rate of variation of 
Sp', and it is clear that we can at the surface change this arbitrarily without changing 
Sp at any point of space. [This cannot be said of the tangential space rates of varia¬ 
tion of Sp, for a change in these causes a change of the same order of magnitude in 
Sp at all points of the surface.] Hence we obtain the equation 
=0.(7), 
or by equation (5), since i — mm s ~ 1 x" 1 b 
[mx {(IbUi' + l ( r V l s — UvSU* r VZ,)}] a _ 6 
= [»ix]« + 6 + 2 V m x}a-b (rVb - iSi T Vl s ) = 0 
The geometrical meaning of equation (7) should be noticed. It reduces the six 
coordinates of the self-conjugate <£/ to three. <£/ operating on any vector reduces it to 
the tangent plane. It may be said to act only on vectors in the plane and to strain 
them m the plane. 
110. We now have 
jjsv {$;} v/ ds = - f(ss Pl 'f - (i'vrv/) ds\ 
