Sir Witt1Am Rowan Hamirton’s Researches respecting Quaternions. 281 
is, therefore, a consequence, and may be considered as an interpretation of the 
very simple algebraical formula for associating three quaternion factors, 
qq 9 =49-99" 
It follows, at the same time, from the theory of cones and conics, that the two 
straight lines, or radu vectores, which are drawn from the origin of coordinates 
to the points R, R,, and which construct the imaginary parts of the two binary 
quaternion annie qq, ¢q', are the two focal lines of a cone of the second 
degree, inscribed in the tetrahedral angle, which has for four conterminous 
edges the four radii which construct the imaginary parts of the three quater- 
nion factors q, 7, q'', and of their continued or ternary product qq‘q" 
48. We have also, by the same associative character of multiplication, an 
analogous formula for the product of any fowr quaternion factors, q, 7’, qq”, 
namely, 
q ‘ gf gf ie = qq’. q f= — q¢ 0." = qs (436) 
if we denote this continued product by q’”; and if we make 
P= FTC HH TT HU, YC HTC e' =4" (437) 
and observe that eee E and F are foci of a spherical conic inscribed in a 
spherical quadrilateral ancp, so that, in the notation recently proposed, 
EF (..) ABCD, (438) 
then also we may write 
FE(..) ABCD, and EF(..) BCDA, (439) 
we shall find, without difficulty, by the help of the formula (435), the five fol- 
lowing geometrical relations, in which each rR is the representative point of the ~ 
corresponding quaternion q : 
een (ai Re es 
Hae) a Bann: 
RR! (..) r/R” RR; (440) 
’ a (ae yar nk R nes 
By Ry ie 3)sRo Re Ra Be 
These five formule establish a remarkable connexion between one spherical 
pentagon and another (when constructed according to the foregoing rules), 
through the medium of five spherical conics ; of which five conics each touches 
two sides of one pentagon, and has its foci at two corners of the other. If we 
2Pp2 
