28 
6. 
ON THE MOTION OF ROTATION OF A SOLID BODY. 
[From the Cambridge Mathematical Journal, vol. ill. (1843), pp. 224—232.] 
In the fifth volume of Liouville’s Journal, in a paper “ Des lois géométriques qui 
régissent les déplacemens d’un système solide,” M. Olinde Rodrigues has given some 
very elegant formulae for determining the position of two sets of rectangular axes with 
respect to each other, employing rational functions of three quantities only. The 
principal object of the present paper is to apply these to the problem of the rotation 
of a solid body ; but I shall first demonstrate the formulae in question, and some others 
connected with the same subject which may be useful on other occasions. 
Let Ax, Ay, Az ; Ax n Ay t , Az t , be any two sets of rectangular axes passing 
through the point A : x, y, z, x n y t , z t , being taken for the points where these lines 
intersect the spherical surface described round the centre A with radius unity. Join 
xx n yy t , zz r by arcs of great circles, and through the central points of these describe 
great circles cutting them at right angles: these are easily seen to intersect in a 
certain point P. Let Px=f, Py=g, Pz = h; then also Px t =/ Py t = g, Pz / = h : and 
ZxPx,= ZyPy,= /-zPz n =6 suppose, 6 being measured from xP towards yP, yP 
towards zP, or zP towards xP. The cosines of f, g, h, are of course connected by the 
equation 
cos 2 /+ cos - g + cos 2 h = 1. 
Let a, /3, 7 ; a', /3', y ; a", /3", y", represent the cosines of x t x, y t x, z,x ; x t y, y,y, z,y ; 
x z, y,z, zz : these quantities are to be determined as functions of /, g, h, 0. 
Suppose for a moment, 
Z yPz = x, Z zPx = y, Z xPy = z ;