Sir WitrrAm Rowan Hamitton’s Researches respecting Quaternions. 245 
ditions included in the formula (219), whatever four numbers may be chosen for 
GsrbvGn Oe 
And if we farther simplify the formule by supposing 
al a On OY CeO aie, (241) 
which will be found in the applications to involve no essential loss of generality, 
we then obtain, from this last-mentioned system of expressions, that system of 
sixty-four numerical values for the sixty-four coefficients of multiplication of 
quaternions, which was assigned in the equations (220), ... (223), of the twenty- 
third article. 
Coefficients of Quaternion- Derivation ; Comparison of Characteristics. 
26. Adopting, then, those values, (220), ...(223), for the sixty-four coefficients 
of multiplication, let us, at the same time, in accordance with the rule (215), 
adopt also such a connected system of values for the sixty-four connected co- 
efficients of derivation, c,,,,, as shall give the continued equation, 
1 = Cy 9 = Con = Con = Cogs = — Cin = Eno = — Cvs 
2 2 
= — Cyop = Cyy3 = Coq = — Cy, = — Cyq3 = —Cgyg = Cyq, = Coy 5 . (242) 
3 
ten of these coefficients c being thus each equal to + 1, and six others being each 
equal to — 1, while the other forty-eight coefficients of derivation shall, by the 
same rule, vanish. 
The formula (125) will thus give the sixteen following equations : 
Dog = ay 3 ay, = ay Ayo — ay 5 ays = a, 3 ] 
Ay = —A5 A, = AH Ay = — A353 AZ = AQ5 { (243) 
Ay) =— 4,5 A) = 8,5 2, = %5 4, = —A,5 | 
A3y = —435 43, = Ans Ay = 85 Ay, = Ay5 
and, therefore, by comparing the definitions (134) and (70), we shall have the 
four expressions : 
id= (ap Ais) i das A) Paavllaye | 
S25 | = (CE cn © es EL) Saale 
xo = (—a, ay ay» —a,) =Jq; 
xq =(—a, —ay a, ay) seca; 
(244) 
