294- Sir W. Rowan Hamilton on Quaternions. 



interest, and might lead to meteorological results of a remark- 

 able nature. 



I cannot conclude this abstract without drawing attention 

 to the circumstance, that M. Arago had already pointed out 

 in 1820, short!}' after CErsted's discovery, the possibility of 

 acting upon the voltaic arc by this magnet, and the analogy 

 which might result between this phasnomenon and that of the 

 aurora borealis. 



XLII. On Qicaternions ; or on a New System of Imaginaries 

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

 M.R.I. A. .f F.Ii.A.S., Correspotiding Member of the Insti- 

 tute of France, Sec., Andrews' Professor of Astronomy in the 

 University of Dublin, and Royal Astronomer of Ireland. 



[Continued from vol. xxxiii. p. 60.] 

 65. TF we make 



p-X=:A^; p— jtA=^^; p—yj=x',', p—fx,'=zij,l; . (115.) 



and in like manner, (see (106.),) 



p-0=-6S = ^,; (116.) 



and if we regard these five newvectors, Apjw,pX/,|«,/,and^pas lines 

 which, being drawn from the centre A, terminate respectively 

 in five new points, l^, Mp l/, m/, and h; while the vector p, 

 drawn from the same centre a, still terminates in the point E, 

 upon the surface of the ellipsoid; then the equations (113.), 

 (114.), of art. 62, will give: 



TK,=Tix.,= Thl = Ti^'=b; . . . (117.) 



while the equations (101.) will enable us to write 



■li^itLili^!hz:h=Y-'o; . . (118.) 



and in like manner, (see (112.),) 



-7- --7- - /_^^ -'^ ^^' • U^9-^ 



this symbol V~^0 denoting (as already explained) a scalar. 

 We shall have also, by (84.), (89.), 



ri^ = A-=\-'0; tZJ!:i=:^^=V-'0; (120.) 



X — » 



the scalars denoted by the symbol V ^0 being not generally 

 obliged to be equal to each other, and being, in these last 



