^44 



substantially the same as mine. This caused me to give up 

 the design of writing on that subject ; and, my thoughts 

 then turning to the theory of light, the subject of surfaces of 

 the second order was also dropped. 



Another form of Theorem I. is given by the property of 

 reciprocal ellipsoids. If a second ellipsoid be constructed, 

 having its centre at O, and its semiaxes coincident with, and 

 inversely proportional to those of the first, and if this ellipsoid 

 be touched by a plane parallel to the invariable plane, it is 

 obvious, from the relations of reciprocal ellipsoids, that the 

 tangent plane will be fixed in space, and that the right line 

 which joins thepoint of contact with the point O, will always 

 be the axis of rotation, and will be proportional to the angu- 

 lar velocity. This form of the theorem, though not men- 

 tioned in the letter, was nevertheless employed in my theory 

 of rotation. It is the form given by M. Poinsot, who uses 

 only the second ellipsoid ; and it has the advantage of de- 

 termining geometrically (as M. Poinsot has remarked) the 

 successive positions of the body in space, independently of 

 the consideration of time ; for the ellipsoid evidently rolls 

 upon the fixed plane which it always touches. This advan- 

 tage, however, though evident when stated, I do not recol- 

 lect that I bad distinctly perceived. 



The theorem mentioned in my letter, for finding the mo- 

 ment of the centrifugal forces, is the same (making allowance 

 for the difference of the ellipsoids) with one given by 

 M, Poinsot, which he speaks of as " a simple and remarkable 

 theorem, containing in itself the whole theory of the I'otation 

 of bodies ;" and of which he further observes, as I have done, 

 that "translated into analysis, it gives immediately the three 

 elegant equations which are due to Euler, and which are 

 usually demonstrated by long circuits of analysis." It was, 

 in fact, from this theorem, by means of the principle of 

 the composition and resolution of rotatory motions, that my 

 theory, as well as that of M. Poinsot, was deduced. I may 



