406 ON THE APPLICATION OF QUATERNIONS TO THE THEORY OF ROTATION. [68
where, putting to abbreviate k = 1 + A 2 + y? + v 2 , we have
tea = 1 + A 2 — ¡j? — v 2 , tea' = 2 (Xy + v) , ko." = 2 (Xv — y) ,
k/3 = 2 (Xy — v) , k/3' — 1 — A 2 + y? — ^ 2 , /c/3 ,/ = 2 ((tty + A) ,
icy = 2 (Xv + y) , icy = 2 (yv — X) , icy" = 1—X* —/j? + v 2 ;
these values satisfying identically the well-known system of equations connecting the
quantities a, /3, y, a', /3', y, a", (3", y".
The quantities a, b, c, 6 being immediately known when A,, /a, v are known, these
last quantities completely determine the direction and magnitude of the rotation, and
may therefore be termed the coordinates of the rotation; A will be the quaternion of
the rotation. I propose here to develope a few of the consequences which may be
deduced from the preceding formulae.
Suppose, in the first place, II = A — 1, then Tb = A — 1, which evidently implies
that the point is on the axis of rotation. The equation IIj = II gives the identical
equations
A (a — 1) + fi /3 +v y =0,
A a' + fx, (/3' — 1) + v y =0,
A cl" + fi (3" +v(y"—l)=0;
from which, by changing the signs of A, p-, v, we derive
A (a — 1) + ¡i a' + v cl" =0,
A ¡3 + /a (/3 7 — 1) + v ¡3" —0,
Ay +/¿7 + v (y" — 1) = 0.
Hence evidently, whatever be the value of n,
AnA“ 1 - II = 0,
if after the multiplication i, j, k are changed into A, ¡i, v, a property which will be
required in the sequel.
By changing the signs of A, /x, v, we also deduce
A -i nA = i (ax + oty + a"z)
+ j + fi'y + P'z)
+ k (yx + yy + y'z),
where a, ¡3, y, a', ¡3', y, a", ¡3", y" are the same as before.
Let the question be proposed to compound two rotations (both axes of rotation
being supposed to pass through the origin). Let L be the first axis, A the quaternion
of rotation, L' the second axis, which is supposed to be fixed in space, so as not to
alter its direction by reason of the first rotation, A' the corresponding quaternion of
rotation. The combined effect is given at once by
n, = A' (AnA“ 1 ) A'- 1 ,
n : = A 7 An (A'A) -1 ;
that is,
