492 Sir W. R. Hamilton on QiiaterniotiSt 



the first as multiplier, and the second as multiplicand ; and 

 therefore was led to introduce quaternions^ such as a-^ib-\-jc 

 ■j-kd, or {a, b, c, d). 



I saw that we had probably / /c = —J, because / k = i ij, and 

 «^ = — 1 ; and that in like manner we might expect to find 

 kj = ijj =r — /; from which I thought it likely that k i :=j, 

 J k = i, because it seemed likely that \{ji =— ij, we should 

 have also kj z=.—jJc, He ■=.— ]< i. And since the order of mul- 

 tiplication of these imaginaries is not indifferent, we cannot in- 

 fer that /^^,or ij ij, is = + 1, because i^ xj^= —1 x — 1 = + 1. 

 It is more likely that k^ = ij ij = — i ijj=—l. And in fact 

 this last assumption is necessary, if we would conform the 

 multiplication of quaternions to the law of the multiplication 

 of moduli. For multiplying a + ib + jc + kd as multiplier 

 into a' -\- i b' -\- j d -\- Jc d' as multiplicand, and granting that 1c^ 

 is real, we find for the real part of the product, a ol — b U —cd 

 + F dd' ; and for the coefficient of ^", ad' + da' -\- terms arising 

 from the multiplication of e andj together; in order then that 

 2aa' dd' may disappear in the expression of the square of the 

 modulus of the product, by being added to 2k-a'u' dd', it is 

 necessary that we should have k'^=—l. 



My assumptions were now completed, namely, 

 ( A.)i2=/2=p_ _ I . ^y_ _ji^j..j]^^ —kj=i', ki= —ik=j: 



and with these I was obliged to assume that if 



{a, b, c, d) (a', b', c', d') = {a", b", c", d"), 

 then the four following equations of multiplication hold good: 



r a" =a a' —b b' — c c' —d d' ; 



. j bl'==ab' + ba' + cd'-dd; 



^ '^ ] c"=ad + ca'-{-db'-bd'; 



^d"=ad' + da' + bc'-cb'. 



But I considered it essential to try whether these equations 



were consistent with the law of moduli, namely, 



fl"2 + 1,112 _,. c"2 + d"^ = {a^ + Z'H c^ + d^) (a'2 + b'^ + d^ + d'% 

 without which consistence being verified, I should have re- 

 garded the whole speculation as a failure. Judge then of 

 my pleasure when, after a careful examination, I found that 

 the whole twenty-four products not involving squares in the 

 development of the sum of the squares of the four quadrino- 

 mials (B.) destroyed each other; and that the sixteen pro- 

 ducts involving s(juares were just those which arise otherwise 

 from the multiplication of a^ + 6^ + c^ + ^^^and a'^ + b'^ + d^^ + d'^^. 



* Mr. Graves has since pointed out to the writer that this theorem of 

 ordinary algebra is not new, and has very elegantly extended it. 



