282 Sir Wiruram Rowan Hamitton’s Researches respecting Quaternions. 
suppose, for simplicity, that each of the ten moduli is = 1, the dependence of six 
quaternions by multiplication on four (as their three binary, two ternary, and one 
quaternary product, all taken without altering the order of succession of the fac- 
tors) will give eighteen distinct equations between the ten amplitudes and the 
twenty polar coordinates of the ten quaternions here considered ; it is therefore 
in general permitted to assume at pleasure twelve of these coordinates, or to 
choose six of the ten points upon the sphere. Not only, therefore, may we in 
general take one of the two pentagons arbitrarily, but also, at the same time, may 
assume one corner of the other pentagon (subject, of course, to exceptional cases) ; 
and, after a suitable choice of the ten amplitudes and four other corners, the five 
relations (440), between the two pentagons and the five conics, will still hold good. 
A very particular (or rather limiting) yet not inelegant case of this theorem 
is furnished by the consideration of the plane and regular pentagon of elementary 
geometry, as compared with that other and interior pentagon which is determined 
by the intersections of its five diagonals. Denoting by k, that corner of the inte- 
rior pentagon which is nearest to the side rr’ of the exterior one; by r’, that 
corner which is nearest to r’ Rr”, and so on to Rr?” ; the relations (440) are satisfied, 
the symbol (..) now denoting that the two points written before it are foci of an 
ordinary (or plane) ellipse, inscribed in the plane quadrilateral, whose corners 
are the four points written after it. We may add, that (in this particular case) 
two points of contact for each of the five quadrilaterals are corners of the interior 
pentagon ; and that the axis major of each of the five inscribed ellipses is equal to 
a side of the exterior figure. 
49. By combining the principles of the forty-seventh with the calculations of 
the twenty-eighth and thirtieth articles, we see that, with the relations (258), 
(259), (284), from which the relations (285) have been already seen to follow, 
we may regard mj, m;, m; as the rectangular coordinates of a point on one focal 
line, and m}>, m>, m3 as the rectangular coordinates of a point on the other focal 
line of a certain cone of the second degree, having its vertex at the origin of those 
coordinates, and having, on the successive intersections of four of its tangent 
planes, four points, of which the coordinates are respectively m,, m.,m,; b, c, d; 
mi, mi, ms; and m;’, mi’, m;'. Hence, with the same relations between the sym- 
bols, the known theory of reciprocal or supplementary cones enables us to infer 
that the two equations 
