202 
Proceedings of Boyal Society of Edinburgh. [sess. 
Before considering the problem in hand — that of finding d.q n 
explicitly — we must consider the properties of a certain linear 
quaternion function of any quaternion. The form of the function 
depends on q and n, and we shall denote it by ( q , n ). Suppose a is 
any quaternion. Split it up into two parts, a' a quaternion co- 
planar with q, and a" a vector perpendicular to the axis of q. There 
are several useful forms of a and a”. Notice that we have 
a' = Sa + component of Va parallel to Yq . 
a " = component of Ya perpendicular to . 
This gives 
a' + a" = a (4), 
a' — Sa + Y^S. Y -1 ^ (5), 
a" = YqVY- 1 qVa = Y- 1 q n YYq n Ya . . . . (6), 
or 2a f> = Y~ 1 q(qa- aq) = Y~ 1 q n (q^a - aq n ) . . . (7). 
These are some of the simple forms of a ' and a ", and we shall 
employ more than one of them. 
The quaternion function (q, n) is defined by the equation 
(q, n)a = nq n ~ l a' + 
• . ( 8 ). 
We will give some of the simpler forms of ( q , n) in full, though 
equation (8) is what we shall use in the present investigation. 
Putting a' = a - a", and substituting for a " the last value in equa- 
tion (7), we get 
(g, = + 'jifa-aif) . . . (9)- 
Again, substituting the first value in equation (6), 
(q, n)a = nqn^a + (V q n - nq n ~ v Vq)YY- l q\a . . (10). 
Another important modification is obtained from the fact that if 
X is a vector perpendicular to the axis of r, 
r\r = TV. A 
(for by Tait’s Quaternions, § 354, r~ x \r = T 2 r.r _2 A). 
n — 1 n — 1 
2 2 a"q 2 = T'-'q.a". 
( 11 ), 
