[ 113] 



XXII. On Quaternions ; or on a New System of Imaginaries in 

 Algebra. By Professor Sir William Rowan Hamilton, 

 LL.D., Corresponding Member of the Institute of France, 

 and Royal Astronomer of Ireland. 



[Continued from p. 31.] 



22. T^HE geometrical considerations of the foregoing article 

 -■- may often suggest algebraical transformations of 

 functions of the new imaginaries which enter into the present 

 theory. Thus, if we meet the function 



aS.u'x" -a'S.a"a, (1.) 



we may see, in the first place, that in the recent notation this 

 function is algebraically a pure imaginary, or vector form, 

 which may be constructed geometrically in this theory by a 

 straight line having length and direction in space; because 

 the three symbols a, a', a" are supposed to be themselves such 

 vector forms, or to admit of being constructed by three such 

 lines; while S.a! a" and S. a." a are, in the same notation, two 

 scalar forms, and denote some two real numbers, positive, ne- 

 gative, or zero. We may therefore equate the proposed func- 

 tion (1.) to a new small Greek letter, accented or unaccented, 

 for example to a!", writing 



ot'" = otS.ct'ct" -u'S.u"u (2.) 



Multiplying this equation by a", and taking the scalar parts 

 of the two members of the product, that is, operating on it by 

 the characteristic S. a" ; and observing that, by the properties 

 of scalars, 



S.«"«S.a'«" = S.«"«.S.a'a» 

 = S. a" «' . S. a" a = S. u" oi S. a" a, 

 in which the notation S.«"aS.a'«" is an abridgement for 

 S (a" a. S. at a"), and the notation S.a" a. S.a' a" is abridged 

 from (S. «" «) . (S. oi a"), while S. a! od' is a symbol equivalent to 

 S(a!a"), and also, by article 20, to S («"«0, or to S. a!' a!, al- 

 though «'a" and a." a! are not themselves equivalent symbols; 

 we are conducted to the equation 



S.a"a'" = 0, (3.) 



which" shows, by comparison with the general equation of -per- 

 pendicularity assigned in the last article, that the new vector 

 a'" is perpendicular to the given vector a", or that these two 

 vector forms represent two rectangular straight lines in space. 

 Again, because the squares of vectors are scalars (being real, 

 though negative numbers), we have 



a(aS.a'a").«' = aVS.a'a" = a'(«S.«'«").«, 



a'(a'S.a"«) .« = «' 2 «S.«"« = a(a'S.a"a).a'; 



Phil. Mag. S. 3. Vol. 29. No. 1 92. August 1 846. I 



