1888 - 89 .] A. M'Aulay on Differentiation of a Quaternion. 203 
n — 1 n—\ 
Again, v a' and q are coplanar q 2 a' q 2 —q n ~ l a'. .*. from eq. (8) 
n ~ l ( YUo w \ — 
(2>»)“ = 2 2 \ na ' + TDg J 2 2 ’ 
Let us here substitute a - a" for a' ; and for a" 
V- W^VVU^Va = i y- 1 U^(U^. a - aUg ra ) . 
Thus 
(q, n)a = q~^~{na + ^(~- a-a\Jq n )}q~ (12). 
This again may be written 
n — 1 7i—l 1/1 
(q, n)a = ?iq 2 aq 2 4 - 1 - — - 
2T^\VU^ YUq 
^(gioKflS-K 2 M) (13). 
We might with ease write a number of other forms. 
It is to be observed that in equations (9), (10), (12), (13) the 
long second term is in every case a vector, for it consists of a vector 
perpendicular to the axis of q operated upon by some quaternion 
which is coplanar with q. Hence 
S{(q,n)a}=nSq n ~ 1 a (14). 
We now proceed to those properties of (q, n) which we require. 
First notice that (q, n) is commutative with any quaternion co- 
planar with q\ and also with (r, m) (which if not sufficiently obvious 
will appear incidentally immediately) if r is a quaternion coplanar 
with q. How we have 
(q, n)(r, m)a = (q, n)(r, m)a + (q, n){r , m)a" 
Yq n Yr m 
- mnq V»- V + by equation (8) 
[ = (r, m)(q, n)a ]. 
Putting then r = q n we get 
(q,n)(q n ,m) = (q,mn) = (q, m)(q m ,n) . . . (15). 
Putting m = - 
° n 
(q, n)(q n , ^ = ( q , 1) = 1 by equations (8) and (4) , 
