432 
Proceedings of the Royal Society 
In the paper it is proved that if we write 
7] = q~ l eq 
£ = 2~‘y2 
(where rj and £ are certain vectors in the body in its initial position) 
the whole kinetic properties of the motion are expressed by the 
equation 
<h = £> 
where <£ is a linear and vector function , which here introduces (as 
the roots of its determining cubic) the three moments of inertia. 
As the tensor of q may have any value whatever, let 
T q = constant. 
Then our equations become 
2»7 = 2 2 , 
72 = i£ , 
H = £■ 
On the integration of these very simple forms the solution of the 
problem depends. They give 
2 <#> _ 1 ( 2 _ 1 72 ) = 2 2 
as the quaternion equation for q; where, however, if forces act, y 
is to be considered as a function of q; and they supply the counter- 
part of Euler’s equations in the form 
cf>r) = — Yr)cf>r ] , 
when y is constant. 
If we seek the actual equations of Euler, referred to the moving 
principal axes, we obtain 
<pk — - Vepe , 
where <p differs from </> simply in the fact that its rectangular unit- 
system is fixed in, and moves with, the body. 
If we write 
q = w + ix+jy -\-Tcz 
the equation above (for q) gives us the following set of ordinary 
