37] 
237 
37. 
ON THE ROTATION OF A SOLID BODY ROUND A FIXED 
POINT. 
[From the Cambridge and Dublin Mathematical Journal, vol. I. (1846), pp. 167—173 
and 264—274.] 
The difficulty of completing elegantly the solution of this problem, in the case 
where no forces act upon the body, arises from the complexity and want of symmetry 
of the ordinary formulae for determining the position of one set of rectangular axes 
with respect to another set; in consequence of which it has hitherto been considered 
necessary to make a particular supposition relative to the position of the fixed axes in 
space, viz. that one of them shall be perpendicular to the “ invariable plane ” of the 
rotating body. But some formulae for the above purpose, given also by Euler, are 
entirely free from these objections. Imagine two sets of axes Ax, Ay, Az, Ax t , Ay t , Az r 
The former set can be made to coincide with the second set, by a rotation 6 round a 
certain axis AR, inclined to Ax, Ay, Az at angles f g, h. (As usual f g, h are the 
angles RAx, RAy, RAz considered as positive, and the rotation is in the same direction 
as a rotation round Az from x towards y.) This axis may be termed the resultant axis, 
and the angle 6 the resultant rotation. The formulae of Euler express the coefficients 
of the transformation in terms of the resultant rotation and of the position of the 
resultant axis, i.e. in terms of 6 and of the angles f g, h, whose cosines are connected 
by the equation 
cos 2 /+ cos 2 <7 + cos 2 h = 1. 
This idea was improved upon by M. Rodrigues (Liouv. tom. v. p. 404), who intro 
duced the quantities 
tan | 6 cos f tan \ 6 cos g, tan \ 6 cos li, 
(quantities which will be represented by g, v) by means of which he expressed the