1890 - 91 .] Mr A. M‘Aulay on Quaternion Differentiation. 109 
This gives us a very simple way of dealing with the external 
couple when the strains are small. For in that case we may put 
m — 1 and V ' = V , when the equations become 
m + 26 = 0 ( 21 ), 
g + -VV € = 0 (22), 
or, instead of the last, 
(8 + iVV®t) + * A=0 (23), 
so that the mathematical problem is the same as if there were no 
external couple, but instead of the given force there were the 
force J5 + JVV S I)L If any part of the given data consist of 
surface tractions, we must add that the given surface traction per 
unit surface must be replaced by g, - JVIVIll, where Uv is 
the unit normal at the point. 
Returning to the consideration of equation (16), we see that if 
the portion considered be limited to the element ds, we obtain 
Sw= -wSSp^Vi (24). 
Before proceeding further, we must consider the strain a little 
more closely. Let x be Professor Tait’s ( Quaternions , 3rd ed., 
§ 384) strain function, so that 
X a>=-Sa>V.p' (25). 
The physical meaning of x ma y be thus explained. Let <*> be the 
vector coordinate before strain of any point P, very near to another 
point 0, relatively to the latter ; then yo> is the coordinate of the 
strained position of P relative to the strained position of 0. In 
symbols 
dp = X d P (26). 
From this equation we may at once deduce the expression for 
V' in terms of V and conversely. For 
S d P V = Stfp'V' = SxdpV' = Stfpx'V' 
. • . since dp is perfectly arbitrary 
V =x V' or V' = x' _1 V ..... (27) 
We have thus from equation (24) 
8w= - mSSp^x " 1 ^ 7 ! (28). 
