* This paper is written wholly from the point of view of mathematical physics, for which a vector algebra is generally admitted to be of supreme importance. The purely analytical aspect of quaternions is not contemplated.

note * page 216 See article “Quaternions” in Chambers's Encyclopaedia (New Edit., 1892).Google Scholar

note * page 217 O'Brien alone of vector analysts uses a non-scalar product of two vectors, which does not involve directly this versorial characteristic; but then he has to introduce his Directrix, so as to get out into space, as it were.

note * page 219 Thus we cannot transform Sin 3θ = ⅓a into Sinθ= .

note * page 221 See also Kelland and Tait's Introduction to Quaternions (chapter iii.) for an even simpler proof.

note * page 222 Macfarlane fully recognises this, and discusses at considerable length his two products of one arrangement of three vectors, and his five products of one arrangement of four vectors. If we represent the vectors of the new system by abed … and the Hamiltonian vectors by αβγδ …, we see at once that ab = − K(αβ) = −jβα. Hence

and the reason why (ab)c and a(bc) have different values is evident. But, by a similar process,

so that, in this non-associative system, each association of one arrangement is equal to a particular association of one other arrangement. In quaternions the different values of the product of three vectors are got by permutation and by permutation only; in this system they are got partly by different associations, and partly by permutation. But each system gives precisely the same number of different products.

Similarly, although there are five different products—((ab)c)d, (a(bc))d, (ab)(cd), a((bc)d), a(b(cd))—got from the one arrangement, any one of these has its four equivalents which are particular associations of other arrangements. Thus it maybe shown that (ab)(cd) = ((ad)b)c = (d(ba))c = b((dc)a) = b(c(ad)). We commend this jungle, which the forsaker of the Hamiltonian track cannot escape if he only go far enough, to the careful consideration of Heaviside. It is small wonder that Grassmann (whose Ausdehnungslehre of 1862 does not hint at the possibility of putting i²= −1) never found leisure to apply his own system to angles in space. He never formed the geometric conception of a quaternion, his vector quotients being quite other things.

† Heaviside regards this name as “ludicrously inefficient,” whatever that may mean. Some name is sorely needed. To invert Delta gives an awkward word, which does not euphoniously combine with other terms; to use Delta is, of course, out of the question. Nabla is certainly enphonious, as any who have used it in lecturing can testify. Being quite innocent of any previous scientific significance, it has just the meaning that is put into it. Nabla occurs only in quaternion analysis; for the operator is not used by Gibbs in its full sense, and that which Heaviside and Macfarlane represent by Δ is not Nabla.

note * page 224 The query suggests itself, could tliese bits have been discovered without quaternions as a guide ?

note * page 225 Tait proves this well-known equation of Poisson by purely quaternionic methods. It is an interesting commentary on the “simplicity” of the positive sign—the fetish alike of Heaviside and Maofarlane—that the quaternion Δ² gives 4π positive on the right-hand side!

note † page 225 Taking his cue from Gibbs, Heaviside might possibly find Ham a less “ludicrously inefficient” name than Nabla for the operator ∇.

note * page 228 Gibbs calls the quantity φ.σ (which is simply Hamilton'sφσ) the direct product of the dyadic φ and the vector σ The direct product of two vectors is α.β(= − Sαβ), and this Heaviside calls the scalar product. Similarly translating the Gibbsian dialect, he speaks of φσ as being the “scalar product of the dyadic and the vector ”—and gets a scalar product equal to a vector! This “i s most tolerable and not to be endured.” Gibbs's own use of direct and of its symbolic “dot ”in two quite different senses is itself open to criticism.

note * page 232 If we except one extraordinary case to be noticed below.