68]
405
68.
ON THE APPLICATION OF QUATERNIONS TO THE THEORY OF
ROTATION.
[From the Philosophical Magazine, vol. xxxm. (1848), pp. 196—200.]
In a paper published in the Philosophical Magazine, February 1845, [20], I showed
how some formulae of M. Olinde Rodrigues relating to the rotation of a solid body
might be expressed in a very simple form by means of Sir W. Hamilton’s theory of
quaternions. The property in question may be thus stated. Suppose a solid body
which revolves through an angle 6 round an axis passing through the origin and
inclined to the axes of coordinates at angles a, b, c. Let
and write
A. = tan \6 cos a, g = tan \6 cos b, v — tan \6 cos c,
A = 1 + i\ ~\~jg T kv \
let x, y, z be the coordinates of a point in the body previous to the rotation,
x x , y x , z x those of the same point after the rotation, and suppose
n = ix +jy + kz ,
II 1 =ix 1 +jy 1 + kz 1 \
then the coordinates after the rotation may be determined by the formula
n x = AnA -1 ;
viz., developing the second side of this equation,
U 1 = i (a x + /3 y + yz )
+j (a! x + /8' y + y' z)
+ k (ol'x + fi"y + y"z),
