^{page 99 note *} With regard to the use of ∇ Professor Tait (Quaternions, 3rd ed., § 149) says :-“The precautions necessary in such matters are twofold—(a) The operator must never be placed after the operand; (b) its commutative or noncommutative character must be carefully kept in view.”

^{page 100 note *} Maxwell's Elect. and Mag., 2nd ed., § 68.

^{page 101 note *} Tait's Quaternions, 3rd ed., § 508. The following words of Professor Tait seem to me to form a powerful argument in favour of my natural notation :- “The highest art is the absence of artifice. … . The difficulties of Physics are sufficiently great in themselves to tax the highest resources of the human intellect; to mix them up with avoidable mathematical difficulties is unreason little short of crime . … . In Quaternions, a subject uniquely adapted to Euclidian space, this entire freedom from artifice and its inevitable complications is the chief feature. … . What is required for Physics is, that we should be enabled at every step to feel instinctively what we are doing. Till we have banished artifice we are not entitled to hope for full success in such an undertaking” (Tait, “On the Importance of Quaternions in Physics,” Phil. Mag., 5th series, vol. xxix. pp. 84–97).

^{page 101 note †} Mess, of Math., vol. xiv. p. 26. I have substituted in the present paper Δ for the ∇ of the original paper.

^{page 102 note *} According to Professor Tait this must be written in some such form as the following:-V∇₁(S∇₂σ₁)σ₂ or S∇₃∇₁. V∇₂σ₁ Sρ₃σ₂ or S∇₃∇₁ S∇₂σ₁. V ρ₃σ₂. Even if the first of these is chosen, it may be pertinently asked why symbols which will, in the nature of things, obey all the laws of vectors, should be restricted in a purely arbitrary manner in obeying those laws. Why should we not be allowed to ring all the possible changes on the form of V∇₁σ₂Sσ₁ as on that of VαβSγδ and reap the corresponding advantages ?

^{page 102 note †} In the Trans. Roy. Soc. Edin., xxvii. p. 251 (1874), Dr Plarr suggests the notation where r is any quaternion. He suggests no notation for where φ is a general linear quaternion function. His symbols ⊲ and ⊳ do not obey the laws of vectors, the first only because it is not allowed the freedom of a vector.

^{page 102 note ‡} The proof is given in the paper already quoted from.

^{page 103 note *} I use the inverted D to suggest the analogy to Hamilton's inverted Δ. It is advisable to write φⱭ, σ∇, &c, instead of Ɑφ, ∇σ, & c, as the numerical suffixes must be much more frequently introduced than the symbols φ or σ which may be very frequently understood, and it is not advisable that both the literal and numerical suffixes should be on the same side of the Ɑ or ∇.

^{page 105 note *} Tait's Quaternions, 3rd ed., §§ 158 et seq.

^{page 105 note †} This frequently-recurring and cumbrous phrase is very annoying. Might I suggest the term Hamiltonian. Thus, in the present case, we should say— “If φ and ψ be two Hamiltonians connected by the equation where Sφζχζ=Sψζχζ where χ is s a perfectly arbitrary Hamiltonian,“&c.

^{page 108 note *} That there is at least very grave doubt whether any such thing as a molecular couple, and therefore that a stress-couple exists (even in the case of magnetism), I hope to show on some future occasion. It is well to learn, however, the nature of the phenomenon, if it possibly exist.

^{page 110 note *} See Tait's Quaternions, 3rd ed., § 381, where it is shown how to determine both ψ and q in terms of χ.

^{page 111 note *} By Tait's Quaternions, 3rd ed., §§ 157, 158, we have V_χμχν=mχ′-1V_μν If μ ν be taken as the conterminous edges of a small parallelogram in the unstrained state V_μν will be its vector area ; χμ, χν will be the edges of the strained parallelogram, and V_χμχν its vector area.

^{page 119 note *} In taking tins for the form of W, and operating upon it as follows, we are following Helmholtz for the particular case when K is a mere scalar. See Wiss. Abh., equation (2d), p. 805. For the various assumptions above see Maxwell's Electricity and Magnetism, part i.

^{page 123 note *} For other particular cases of the stress under consideration, see Helmholtz, Wiss. Abh., i. 798 ; Korteweg, Wied. Ann., ix. 48; Lorberg, Wied. Ann., xxi. 300 ; Kirehoff, Wied. Ann., xxiv. 52, xxv. 601.