Mathematical Notes by Professor Tait - 3. On some Quaternion Transformations

Extract

Since the algebraic operator

when applied to any function of x, simply changes x into x + h, it is obvious that if σ be a vector not acted on by

we have

whatever function f may be.

If Δ bear to the constituents of σ the same relation as ∇ bears to those of ρ, and if f and F be any two functions which satisfy the commutative law in multiplication, this theorem takes the curious form

of which a particular case is

The modifications which the general expression undergoes, when fand F are not commutative, are easily seen and need not be indicated in this abstract.