THE 

 LONDON, EDINJ^URGH and DtJBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



[THIRD SERIES.] 



OCTOBER 1844. 



XXXIX. On Qiiaternions ; or on a new System of Imaginaries 

 in Algebra. By Sir William Rowan Hamilton, LL.D., 

 P.RJ.A., F.R.A.S., Ho7i. M. R. Soc. Ed. and Dub., Hon. 

 or Corr. M. of the Royal or Imperial Academies of St. Pe- 

 tersburgh, Berlin, Turi?i, and Paris, Member of the American 

 Academy of Arts and Scietices, and of other Sciejitifc Socie- 

 ties at Home and Abroad, Andrews' Prof, of Astrojiomy in 

 the Utiiversity of Dublin, and Royal Astronomer of Ireland. 

 [Continued from p. 13.] 



6. TT follows immediately from the principles already ex- 

 -■- plained, that if R R' R" be any spherical triangle, and 

 if a /3 y be the rectangular co-ordinates of R, «' /3' y' of R', and 

 a" /3" y' of R", the centre O of the sphere being origin and the 

 radius unity, and the positive semiaxis of z being so chosen as 

 to lie to the right or left of the positive semiaxis of 3/, with re- 

 spect to the positive semiaxis of x, according as the radius 

 OR" lies to the right or left of OR' with respect to OR, then 

 the following imaginary or symbolicybr»/z//a of midtiplication 

 of quaternions will hold good : 



(cos R + {ia+j^ + ky)sm R } {cos R' + {ia' +J^' + kry')sm R'} 

 = -cosR"+(m"+i^"+^7")sinR"; .... (I.) 



the squares and products of the three imaginary units, i,j, /c, 

 being determined by the nine equations of definition, assigned 

 in a former article, namely, 



e2 =/ = A:2 = — 1 ; (A.) 



ij=i/c,jk=s:i,Ici=j; (B.) 



ji=-k, kj=-i, ik--j. .... (C.) 



Developing and decomposing the imaginary formula (I.) by 

 these conditions, it resolves itself into the four following real 

 equations of spherical trigonometry : 



Phil. Mag. S. 3. Vol. %^. No. 166. Oct. 1844. R 



