298] 
CERTAIN SPECIAL PROBLEMS OF DYNAMICS. 
559 
so that representing the right-hand side by 
W + iX+jY+kZ, 
we have identically 
W 2 4- X 2 + Y 2 + Z 2 = (w 2 + or? + y 2 + z 2 ) (iu 2 + x 2 + y 2 + z' 2 ). 
It is hardly necessary to remark that Sir W. R. Hamilton in his various publications 
on the subject, and in the Lectures on Quaternions, Dublin, 1853, has developed the 
theory in detail, and has made the most interesting applications of it to geometrical 
and dynamical questions ; and although the first explicit application of it to the 
present question may have been made in my own paper next referred to, it seems 
clear that the whole theory was in its original conception intimately connected with 
the notion of rotation. 
141. Cayley, “On certain Results relating to Quaternions” (1845).—It is shown 
that Rodrigues’ transformation formula may be expressed in a very simple manner by 
means of quaternions ; viz., we have 
ix + jy + kz = (1 + i\ +jfi + lev)- 1 (iX +jY + JeZ) (1 + i\ +j/j, + kv), 
where developing the function on the right-hand side, and equating the coefficients of 
i, j, k, we obtain the formulae in question. A subsequent paper, Cayley, “ On the 
application of Quaternions to the Theory of Rotation ” (1848), relates to the composition 
of rotations. 
Principal Axes, and Moments of Inertia. Article Nos. 142—163. 
142. The theorem of principal axes consists herein, that at any point of a solid 
body there exists a system of axes Ox, Oy, Oz, such that 
But this, the original form of the theorem, is a mere deduction from a general theory 
of the representation of the integrals 
for any axes through the given origin by means of an ellipsoid depending on the 
values of these integrals corresponding to a given set of rectangular axes through the 
same origin. 
143. If, for convenience, we write as follows, M = I dm the mass of the body, and