1890-91.] Mr A. M‘Aulay on Quaternion Differentiation. Ill 
where tsr is a self-conjugate linear vector function of a vector defined 
by the equation 
tfw = q~ l <f>(c[ioq~ x )q (33). 
Substituting for respectively in the last expression for 
Sir, we finally get 
8w = - mSSif/iiTffif/^Z (34). 
Thus the only infinitesimal appearing in Sw is Si f/. It follows that 
w can only be a function of the pure strain if/. This is not an 
obvious truth, as I have before remarked. Assuming it now, we 
have by equation (13) 
Sw = - SSi/^aw;£ . 
Comparing this with the last equation, and noticing that Sif/ is a 
perfectly arbitrary self-conjugate linear vector function of a vector, 
we see (p. 103 above) that 
2 v |,aw = m( 2 ffi/f _1 + i/'~V) (35), 
V J(srt /'" 1 + is the pure part of 73 * 1 if/' 1 . The last equation 
can be easily shown to lead to 
mzsoj - ^dwxl/o) + Y Oif/u) . . . \ 
where r . . . (36). 
e = (xf/ + sm)- 1 ^aw^ .) 
This, with equation (33) above, completes the present problem of 
expressing the stress <£ explicitly in terms of the strain q and if/. 
7v can be easily shown to be that stress which cf> becomes when 
the body is rotated by the operator g _1 ( )q, without altering the 
force exerted across any interface of the body. We thus see why 
it is ts and not <f> which bears the simplest relation to if/. 
<f)Q) or (f>(jy + Yew, where e is perfectly arbitrary so far as the strain 
is concerned, is the force exerted on the strained area w. This last 
would more properly be denoted by w'. Calling it w', and denot- 
ing by w the same vector area before strain, we have *■ 
(o' = (37). 
* By Tait’s Quaternions , 3rd ed., §§ 157, 158, we have Y xt JL X v==m X “W fxv. 
If fx , v be taken as the conterminous edges of a small parallelogram in the 
unstrained state Y/xv will be its vector area ; x ^ X v will be the edges of 
the strained parallelogram, and Vxv-X 1 ' its vector area. 
