1890-91.] Mr A. M‘Aulay on Quaternion Differentiation. 115 
Since we are dealing with small strains, it is convenient to alter 
the notation slightly. Instead of our previous y and ifr we shall 
now write 1 + y anc ^ 1 + ^ respectively, so that both y and if/ are 
small, and if/ is the pure part of y. And, further, w'e shall write 
p = p + 7], so that rj is the small displacement. Thus 
yoD = — Sa> V .7] (16) 
ifru) = — ^(^Sw V 1 + V .... (17). 
Since the strain ifr is small, the stress z*r, i.e., Qw is linear in if/, 
and .’. w is quadratic. Now, for any such quadratic expression as 
can be easily proved 
w= -|Si fs£ffUo£ (18), 
. \ from equation (45) we have 
M? = - JSi/r£ar£ (19). 
Also w, being quadratic in if/, is, if regarded as a function of zsr, 
quadratic in it. Hence 
-|S (50), 
and .'.by the last equation and p. 104 above, 
^=„aw (51). 
Instead of regarding w as a function of if/ or of nr, it is perhaps 
simpler, from the mathematical point of view, to start with assum- 
ing it a given function of the first space derivatives of rj. Let, in 
fact, 
w = w(v 1 ,rj 1 ,v 2 ,rj 2 ) ( 52 ) 
where w(a,/3,y,8) is a scalar function of the vectors a, ft , y, S, which 
is (1) linear in each of its constituents ; (2) symmetrical in a and (3, 
and also in y and 8 ; (3) such that the pair a, ft may be interchanged 
with the pair y, 8 without altering the value of the function. (If 
this last is not true in the first form of w chosen, it may be made 
so by writing \iv(a,ft,y,8) + ^w(y,8,a,/3) instead of w(a,/3,y,8), as this 
does not affect equation (52).) Such a function can be proved to 
involve twenty-one independent scalar constants, which is the number 
also required to determine an arbitrary homogeneous quadratic 
function of the six coordinates of ifr. 
