Sir W. Rowan Hamilton on Quaternions. 2g9 



R' R Ry being the spherical angle at R, measiu'ed from R R' 

 to R Rp and similarly in other cases. But these equations 

 between the spherical angles of the figure are precisely those 

 which are requisite in order that the two points R, and R^^ 

 should be the two foci of a spherical co?iic inscribed in the sphe- 

 rical quadrilateral R R' R" R'", or touched by the four great 

 circles of which the arcs R R', R' R", R" R'", R'" R are parts ; 

 this geometrical relation between the six representative points 

 R R' R" R^ R,^ R"' of the six quaternions Q, Q', Q", QQ', 

 Q'Q", QQ'Q", which may conveniently be thus denoted, 



R^ R/^ ( . . ) R R' R" R"'> .... (Q".) 

 is therefore a consequence, and may be considered as an in- 

 terpretation, of the very simple algebraical theorem for three 

 quaternion factors, 



QQ'.Q" = Q.Q'Q" (Q.) 



It follows at the same time, from the theory of spherical co- 

 nies, that the two straight lines, or radii vectores, which are 

 drawn from the origin of coordinates to the points Ry, R^,, 

 and which construct the imaginarij parts of the two binary qua- 

 ternion products Q Q', Q' Q", are the two focal lines of a cone 

 of the second degree, inscribed in the pyramid which has for its 

 four edges the four radii which construct the imaginary parts of 

 the three quaternion factors Q, Q', Q", and of their continued 

 {or ternary) -product Q Q' Q". 



16. We had also, by the same associative character of mul- 

 tiplication, analogous formulae for any four independent fac- 

 tors, 



Q . Q' Q" Q'" = Q Q' . Q" Q'" = &c. ; . . (Q'.) 

 if then we denote this continued product by Q'^, and make 

 QQ' = Q^, Q/Q'' = Q/, Q//QW=:Q;', 

 QQ'Q" = q;", Q'Q"Q'"=Q/v, 



and observe that whenever E and F are foci of a spherical 

 conic inscribed in a spherical quadrilateral A B C D, so that, 

 in the notation recently proposed, 



E F ( . . ) A B C D, 

 then also we may write 



FE(..) ABCD, and EF(..)BCDA, 



we shall find, without difficulty, by the help of the formula 

 (Q".), the five following geometrical relations, in which each 

 R is the representative point of the corresponding quaternion 

 Q: 



