306 Sir W. Rowan Hamilton on Quaternions. 



Theorems will hold good, according as the number n is odd 

 or even. 



Theorem I. If n be odd, and if we draw two tangent -planes 

 to the surface at the points p„, p.2„, meeting the two new chords, 

 PP2n> Pn P3«» respectively, in two new points, n, r' ; then the 

 three points brr' shall be situated on one straight line. 



Theorem II. If ;i be even, and if we describe two pairs of 

 plane conies on the surface, each conic being determined by 

 the condition of passing through three points thereon, as follows: 

 the first pair of conies passing through bpp2m, and P„P2nP3n; 

 and the second pair through BP^Pg^ and pp^Pg^; it will then 

 be possible to trace, on the same surface, two other plane conies, 

 of which the first shall touch the two conies of the first pair, at 

 the two points b and p„; while the second new conic shall touch 

 the two conies of the second pair, at the two points b and Pg^. 



90. With respect to the^rs^of the two theorems thus com- 

 municated, it may be noticed now, that it gives an easy mode 

 of resolving the following Problem, analogous to a celebrated 

 problem in plane conies : — To find the two (real or imaginary) 

 polygons, BBjBg . . B»_i and b'b'iB'q . . b'„_i, with any given odd 

 number n of sides, which can be inscribed in a given surface 

 of the second order, so that their n successive sides, namely 

 BB], BiBgj • • for one polygon, and b'b'j, b'iB'q, . . for the other 

 polygon thus inscribed, shall pass respectively through w given 

 points Aj, Ag, . . An, which are not themselves situated upon 

 the surface. For we have only to assume at pleasure any 

 point p upon that surface, and to deduce thence the two non- 

 superficial points lately called r and r', by the construction 

 assigned in the theorem ; since by then joining the two points 

 thus found, the joining line rr' will cut the given surface qf the 

 second order in the two (real or imaginary) points, b, b', which 

 are adapted to be, respectively, the first corners of the two poly- 

 gons required. — That there are (in general) two such (real or 

 imaginary) polygons, when the number of sides is odd, had been 

 previously inferred by the writer, from the quaternion analysis 

 which he employed. Indeed, it may have been perceived to 

 be, through geometrical deformation, a consequence of what 

 was stated in § XIV. of article 87 of this series of papers on 

 Quaternions, for the particular case of the ellipsoid, in the 

 Philosophical Magazine for September 1849. See also the 

 account, in the Proceedings of the Royal Irish Academy, of 

 the author's communication to that body, at the meeting of 

 June 25th, 184'9; in which account, indeed, will be found 

 (among many others) both the theorems of the preceding 

 article 89 ; the second of those theorems being however there 

 enunciated under a metric, rather than under a graphic form. 

 [To be continued.] 



