ON SOME EXTENSIONS OF QUATERNIONS By

William Rowan Hamilton

(Proceedings of the Royal Irish Academy, 6 (1858), pp. 114–115)

Edited by David R. Wilkins

On some Extensions of Quaternions.

Sir William Rowan Hamilton.

Read June 26th, 1854.

[Proceedings of the Royal Irish Academy, vol. vi (1858), pp. 114–115.]

Sir W. R. Hamilton read a paper on some extensions of quaternions:

Besides some general remarks on associative polynomes, and on some extensions of the modular property, Sir W. R. Hamilton remarked that if, in the quadrinomial expression

Q = w + ιx + κy + λz,

the laws of the symbols ι κ λ be determined by the following formula of vector-multiplication,

(A) … (ιx + κy + λz)(ιx′ + κy′ + λz′)
= (m_{1}^{2} − l_{2}l_{3})xx′ + (l_{1}m_{1} − m_{2}m_{3})(yz′ + zy′)
+ (m_{2}^{2} − l_{3}l_{1})yy′ + (l_{2}m_{2} − m_{3}m_{1})(zx′ + xz′)
+ (m_{3}^{2} − l_{1}l_{2})zz′ + (l_{3}m_{3} − m_{1}m_{2})(xy′ + yx′)
+ (ιl_{1} + κm_{3} + λm_{2})(yz′ − zy′)
+ (κl_{2} + λm_{1} + ιm_{3})(zx′ − xz′)
+ (λl_{3} + ιm_{2} + κm_{1})(xy′ − yx′),

then this expression, which he proposes to call a quadrinome, has many properties (associative, modular, and others), analogous to the quaternions; which latter are indeed only that case of such quadrinomes, for which,

l_{1} = l_{2} = l_{3} = 1, m_{1} = m_{2} = m_{3} = 0, ι = i, κ = j, λ = k.

He has, however, found another distinct sort of associative quadrinomial expression, which has also several analogous properties, and for which he suggests the name of tetrads; the product of two vectors being in it,

\begin{aligned}(B) … & (lx + my + nz)(lx′ + my′ + nz′) \\ & + (κn − λm)(yz′ − zy′) + (λl − ιn)(zx′ − xz′) + (ιm − κl)(xy′ − yx′). \end{aligned}