1890-91.] Mr A. M'Aulay on Quaternion Differentiation . 117 
where m stands for c + ~ . This last may be derived in a great 
O 
many other ways still more simple. 
The equation of equilibrium thus becomes 
g = rcV 2 77 + mVSV77 (63). 
As a final application in the theory of elasticity let us consider 
St Tenant’s torsion problem from a quaternion point of view. Let 
r, <f>, z be ordinary columnar coordinates whose axis is parallel 
to the generating lines of the cylinder. Let A, fi, v be unit vectors 
in the directions of dr , dcfj, dz respectively, so that 
_ . d ad d 
V = A.— H — + v— • 
dr r d<J> dz 
It is required to determine v a scalar function of r and <£, so that rj 
given by the equation 
rj = r(zrfji + vv) . (64)) 
where r is a given small scalar constant, may satisfy (1) equation 
(63) of equilibrium, and (2) the equation 
0 = = - wSw V . r] — n V iSco^ — (m — w)o>S V rj 
at every point of the curved surface ; co in this case standing for 
the normal at the point. Q having the same general meaning as 
on p. 104 above, we have 
Q( v 1 ^ 1 ) = T MQ( V) ~ Q(^A)] + + Q( V v,v)} (65). 
In the case when Q is symmetrical in its constituents this takes the 
simple form 
Q( ^ 7 D'7i) = 'r{^Q(^) + Q( Vv,v)} .... (66). 
From the first of these we have 
V rj = r{ (2 zv — rX) 4- V w } 
= r{ V ( t ? — ^r 2 ) + Vw} 
so that SV ?7 = 0 and V 2 r] = vV 2 v. Thus equation (63) becomes 
V 2 v = 0 (67). 
From equations (62) and (66) above, we see that in the present 
case the stress is given by 
7 X(d = - nr{r(fji$(i)v + vStoju,) + (vSto Vv + V^Scov)} . (68), 
