203 Sir W. Rowan Hamilton on Quaternions. 



the Cartesian Coordinates, or by some less algebraical and 

 more purely geometrical method, the following theorems (if 

 not already known), which have thus been Jbwid by the Qua- 

 ternions, will doubtless be led to perceive ma7ii/ new truths, 

 connected with them, which have escaped the present writer ; 

 although he too has arrived at other connected results, which 

 he must suppress in the following notice. 



87. I. An ellipsoid (e) being given, and also a system of any 

 even number of points of space, Aj, Ag, .. A^mi of which points 

 it is here supposed that none are situated on the surface of the 

 ellipsoid ; it is, in general, possible to iiiscribe in this ellipsoid, 

 two, and only iwo, distinct and real polygons of 2m sides, 

 BBj . . B^OT-i and b'b'i . . B'2,ra_i, such that the sides of each of 

 these two polygons (b)(b') shall pass, respectively and success- 

 ively, through the 2m given points ; or in other words, so that 

 BAjBi, B1A2B2, ... B2„j_iA2mB, and also b'AjB'i, B'jAgB'g, ... b'2^_, 

 A2otB', shall be straight lines; while b, Bj, ... B2m-i) and also 

 b', b'j, ... b'2to-ij shall be points upon the surface of the ellip- 

 soid. 



[It should be noted that there are also, in general, what 

 may, by the use of a known phraseology, be called two other, 

 but geometrically^ imaginary, modes of inscribing a polygon, 

 under the same conditions, in an ellipsoid: which modes may 

 become real, by imaginary deformation, in passing to another 

 surface of the second order.] 



II. If we now take any other and variable point p on the 

 ellipsoid (e) instead of b or b', and make it the first cotmer of 

 an inscribed polygon of 2m + 1 sides, of which the frst 2m 

 sides shall pass, respectively and successively, through the 2m 

 given points (a); in such a manner that pAjP^, PiA^Pj* ••• 

 P2OT-1 A2,„ P2„i, shall be straight lines, while p, Pj, Pg, ... v^m 

 shall all be points on the surface of the ellipsoid : then the last 

 side, or closing chord, P2toP> of this new and variable polygon 

 (p), thus inscribed in the ellipsoid (e), shall touch, in all its 

 positions, a certain other ellipsoid (e'). 



III. This new ellipsoid (e') is itself inscribed in the given 

 ellipsoid (e), having double contact therewith, but being else- 

 where interior thereto. 



IV. The two points of contact of these two ellipsoids are the 

 points B and b'; that is, they are ihe first corners of the two 

 inscribed polygons of 2m sides, (b) and (b'), which were con- 

 sidered in I. 



[So far, the results are evidently analogous to known theo- 

 rems, respecting polygons in conies; what follows is more 

 peculiar to space.] 



V. If the two ellipsoids, (e) and (e'), be cut by any plane 



