TALBOT3858

Lacock Abbey Chipp^m
Wilts – Lacock Abbey

Kell^d

9 Jan^y 63

Thanks for y^r explanⁿ – In the quarternⁿ system, some of the rules of common Algebra are subverted & others not, so that a beginner is easily perplexed –

On looking at Hamiltons book, I found his demonstⁿ that αβγ = γβα (these being vectors) and also he gives also the following extended theorem

If αβγδε &c be any odd n^o of vectors their product may be inverted so that α·β·γ·δ·ε = ε·δ·γ·β·α And But if their n^o be even, this eqⁿ is not true. And on consider^g this theorem I found out a v^y simple demonstratⁿ of it for myself, w^ch I think must be correct.

Can you inform me whether the follow^g theorem is correct, w^ch I deduce f^m the quaternⁿ doctrines? It looks like one of the ordinary theorems about coordinates, but I have got no books here, & my memory fails me on the subject.

Theorem. If O <illegible deletion> OX OY OZ are 3 rectang^r axes of coordinates, & OP, OQ are 2 given lines fixed in positⁿ issuing from the origin O – and if a b c are the project^ns of OP on the 3 axes, & a′b′c′ those of OQ Then aa′ + bb′ + cc′ is a cons^t quant^y, however the situatⁿ of the axes is changed.

<expanded version>

Lacock Abbey Chippenham
Wiltshire – Lacock Abbey

Kelland

9 January 1863

Thanks for your explanation – In the quarternion system, ¹ some of the rules of common Algebra are subverted & others not, so that a beginner is easily perplexed –

On looking at Hamiltons book, ² I found his demonstration that αβγ = γβα (these being vectors) and also he gives also the following extended theorem

If αβγδε &c be any odd number of vectors their product may be inverted so that α·β·γ·δ·ε = ε·δ·γ·β·α And But if their number be even, this equation is not true. And on considering this theorem I found out a very simple demonstration of it for myself, which I think must be correct.

Can you inform me whether the following theorem is correct, which I deduce from the quaternion doctrines? It looks like one of the ordinary theorems about coordinates, but I have got no books here, & my memory fails me on the subject.

Theorem. If O <illegible deletion> OX OY OZ are 3 rectangular axes of coordinates, & OP, OQ are 2 given lines fixed in position issuing from the origin O – and if a b c are the projections of OP on the 3 axes, & a′b′c′ those of OQ Then aa′ + bb′ + cc′ is a constant quantity, however the situation of the axes is changed.