222 Sir W. Rowan Hamilton on Quaternions. 



(cos Ri + /rj sin Ri) (cos Rg + i^^ sin Rg)"! , ^ . 



...(cosR„+/R„sinRJ = (-l)"/' ' ' ^ 'f 

 14<. For the case of a spherical triangle R R' R", this rela- 

 tion becomes 

 (cos R + /r sin R) (cos R' + «R/sin R') (cos R" + ?R//sin R")l ,„ , 



= -1; J ^ '^ 



and reproduces the formula (I'.), when we multiply each mem- 

 ber, as multiplier, into cos R" — zr " sin R" as multiplicand. 

 The restriction, mentioned in a former article, on the direc- 

 tion of the positive semiaxis of one coordinate after those of 

 the two other coordinates had been chostn, was designed 

 merely to enable us to consider the three angles of the tri- 

 angle as being each positive and less than two right angles, 

 according to the usage commonly adopted by writers on sphe- 

 rical trigonometry. It would not have been difficult to deduce 

 reciprocally the theorem (R.) for any spherical polygon, from 

 the less general relation (!'.) or (F.) for the case of a spherical 

 triangle, by assuming any point P upon the spherical surface 

 as the common vertex of ti triangles which have the sides of 

 the polygon for their n bases, and by employing the associa- 

 tive character of multiplication, together with the principle 

 that codirectional quaternions, when their moduli are supposed 

 each equal to unity, are multiplied by adding their amplitudes. 

 This last principle gives also, as a verification of the formula 

 (R.), for the case of an infinitely small, or in other words, a 

 plane polygon, the known equations, 



cosSR = (-l)'!, sinSR = 0. 

 15. The associative character of multiplication, or the for- 

 mula (Q.), shows that if we assume any three quaternions Q, 

 Q', Q", and derive two others Q^, Q^^ from them, by the equa- 

 tions 



QQ' = Qy, Q'Q" = Q/„ 



we shall have also the equations 



Q,Q" = QQ, = Q'", 

 Q'" being a third derived quaternion, namely the ternary pro- 

 duct Q Q' Q". Let R R' R" R, R^, R'" be the six representa- 

 tive points of these six quaternions, on the same spheric sur- 

 face as before ; then, by the general construction of a product 

 assigned in a former article*, we shall have the following ex- 

 pressions for the six amplitudes of the same six quaternions: 

 6 = R'RR/ =R,/RR'"; fi^=R"R,R'" =7r-RR^R'; 

 6' = R^R'R =R"R'R^,; 9^/=R"'R^,R =7r-R'R^^R"; 

 fl"= R^^ R" R' - R'" R" R^ ; 6'" =,r- R^ R'" R"=:,r- R R"' R,^ ; 

 * In the Number of this Magazine for July 1844, S. 3. vol. xxv. 



