221 



then the quadratic equation becomes 

 q* = qi + j : 

 and two unequal roots of it are the following : 



q, = 1(1 + i + j - k), 



?» = K - l + * -J ~ h )' 

 Substitution and reduction give hence these two expressions : 



2mr 



J 







i + J . . 2mr , . (2» - 1)tt ' 

 z sin— k sin - 



\an-i . (2w - 1) 7T 

 2 h- - — sin- _ J 



i + 



= 1 - 



t - k ' . 2(« - 1) 7T . . 2?«7T 



srn -i — + ,; sm — 



which may easily be verified by assigning particular values to 

 n. No importance is attached by the writer to these particular 

 results: they are merely offered as examples. 



5. It may have appeared strange that Sir William R. Ha- 

 milton should have spoken of two unequal quaternions, as being 

 among the roots, or two of the roots, of a quadratic equation in 

 quaternions. Yet it was one of the earliest results of that cal- 

 culus, respecting which he made (in November, 1843) his ear- 

 liest communication to the Academy, that such a quadratic 

 equation (if of the above-written form) has generally six roots: 

 whereof, however, two only are real quaternions, while the 

 other four may, by a very natural and analogical extension of 

 received language, be called imaginary quaternions. But the 

 theory of such imaginary, or ■partially imaginary quaternions, 

 in short, the theory of what Sir William R. Hamilton has 

 ventured to name " Biquaternions," in a paper already pub- 

 lished, appears to him to deserve to be the subject of a sepa- 

 rate communication to the Academy. 



