Sir Witt1am Rowan Hamivton’s Researches respecting Quaternions. 285 
Note A. 
Extract from a Letter of Sir William R. Hamilton to John T. Graves, Esq. 
“ Observatory of Trinity College, Dublin, 24th October, 1843. 
“ The Germans often putz for / —1, and therefore denote an ordinary imaginary 
quantity by « + zy. I assume ¢hree imaginary characteristics or units, 7,7, k, such that 
each shall have its square = — 1, without any one being the equal or the negative of any 
other ; 
t—j7— hs — 118 (1) 
And I assume (for reasons explained in my first letter) the relations 
y= ks WSR kt=); (2) 
S—h; VijS—15) tha=7; (3) 
each imaginary unit being thus the product of the two which precede it in the cyclical 
order ijk, but the negative of the product of the two which follow it in that order. Such 
being my fundamental assumptions, which include (as you perceive) the somewhat strange 
one that the order of multiplication of quaternions is not, in general, indifferent, 1 have 
at once the theorem that 
(w+ ix +jy +hz) (w! +x! +)y' +k2') = w" +n! +jy” +hz", (4) 
if the following relations hold good : 
ww” =ww' —xx' —yy —z2'; (5) 
v= wa! +aw' +y2/—zy'; 
yl awy +yw! +20! — 22’; (6) 
2 = w2! +20! + 2y' — ya’ ; 
and reciprocally that these four relations (5) and (6) are necessary (on account of the 
mutual independence of the three imaginary units, 7, 7, k, except so far as they are con- 
nected by the conditions above assigned), in order that the quaternion w’ +ix” +jy +h2”’ 
may result as a product from the multiplication of w’ + ix’ +jy’+hz’, as a multiplicand, by 
w+iz+jy+hz as a multiplier. 
“¢ Making, for abridgment, 
z/=wal +20; yy =wy'+yw'; 2//=w2'+zw'; (7) 
a! =y2—2y! sy, = 2a —ax's 2,/=2y'—ya'; (8) 
and observing that 
Ey HYY/ +2z,/" = 0, xx, + yy, a6 Zip = 0 3 (9) 
we see easily that 
By EF YLY 2,12, =0 3 (10) 
