408 ON THE APPLICATION OF QUATERNIONS TO THE THEORY OF ROTATION. [68 
For two axes L, L' fixed in space, 
U, = AAH (A'A)- 1 - (AT'A'" 1 - T) - A' (ATA -1 - T) A' -1 ; 
and so on for any number, the last terms being always a translation. If the two axes 
are parallel, and the rotations equal and opposite, 
A = A'" 1 , 
whence 
n x = n + A' (r — F) A'“ 1 (r - P) ; 
or there is only a translation. The constant term vanishes if i, j, k are changed into 
A', /j!, v, which proves that the translation is in a plane perpendicular to the axes. 
Any motion of a solid body being represented by a rotation and a translation, it 
may be required to resolve this into two rotations. We have 
n x = AjIIAj -1 + T, 
where T is a given quaternion whose constant term vanishes. Hence, comparing this 
with the general formula just given for the combination of two rotations, 
Aj — il^A'A, 
T = - (AT'A'“ 1 - P) - A' (ATA- 1 - T) A' -1 , 
the second of which equations may be simplified by putting A /-1 TA' = $, by which 
it may be reduced to 
8 = (A'-T'A' - F) - (ATA- 1 - T), 
which, with the preceding equation A x = J^A'A, contains the solution of the problem. 
Thus if A or A' be given, the other is immediately known; hence also S is known. 
If in the last equation, after the multiplication is completely effected, we change 
i, j, k into A, /a, v, or A', ¡x, v, we have respectively, 
S = A'-T'A' — F, $ = — (AHA -1 — T), 
which are equations which must be satisfied by the coefficients of F and T respectively. 
Thus if the direction of one axis is given, that of the other is known, and the axes 
must lie in certain known planes. If the position of one of the axes in its plane be 
assumed, the equation containing S divides itself into three others (equivalent to two 
independent equations) for the determination of the position in its plane of the other 
axis. If the axes are parallel, A, ¡x, v are proportional to A', ¡x, v ; or changing 
i, j, k into A, fi, v, or A', /¿', v, we have S= 0; or what is the same thing, T = 0, which 
shows that the translation must be perpendicular to the plane of the two axes. 
If p, q, r have their ordinary signification in the theory of rotation, then from 
the values in the paper in the Cambridge Mathematical Journal already quoted, 
«(ip+jq + kr) = 2-^A + ^j