I Introduction

Quaternions as a practical instrument of physical research

I
Introduction

It is a curious phenomenon in the History of Mathematics that the greatest work of the greatest Mathematician of the century which prides itself upon being the most enlightened the world has yet seen, has suﬀered the most chilling neglect.

The cause of this is not at ﬁrst sight obvious. We have here little to do with the beneﬁt provided by Quaternions to Pure Mathematics. The reason for the neglect here may be that Hamilton himself has developed the Science to such an extent as to make successors an impossibility. One cannot however resist a strong suspicion that were the subject even studied we should hear more from Pure Mathematicians, of Hamilton’s valuable results. This reason at any rate cannot be assigned for the neglect of the Physical side of Quaternions. Hamilton has done but little in this ﬁeld, and yet when we ask what Mathematical Physicists have been tempted by the bait to win easy laurels (to put the incentive on no higher grounds), the answer must be scarcely one. Prof. Tait is the grand exception to this. But well-known Physicist though he be, his fellow-workers for the most part render themselves incapable of appreciating his valuable services by studying the subject if at all only as dilettanti. The number who read a small amount in Quaternions is by no means small, but those who get further than what is recommended by Maxwell as imperatively necessary are but a small percentage of the whole.

I cannot help thinking that this state of aﬀairs is owing chieﬂy to a prejudice. This prejudice is well seen in Maxwell’s well-known statement—“I am convinced that the introduction of the ideas, as distinguished from the operations and methods of Quaternions, will be of great use to us in all parts of our subject.”² Now what I hold and what the main object of this essay is to prove is that the “operations and methods” of Quaternions are as much better qualiﬁed to deal with Physics than the ordinary ones as are the “ideas”.

But, what has produced this notion, that the subject of Quaternions is only a pretty toy that has nothing to do with the serious work of practical Physics? It must be the fact that it has hitherto produced few results that appeal strongly to Physicists. This I acknowledge, but that the deduction is correct I strongly disbelieve. As well might an instrument of which nobody has attempted to master the principles be blamed for not being of much use. Workers naturally ﬁnd themselves while still inexperienced in the use of Quaternions incapable of clearly thinking through them and of making them do the work of Cartesian Geometry, and they conclude that Quaternions do not provide suitable treatment for what they have in hand. The fact is that the subject requires a slight development in order readily to apply to the practical consideration of most physical subjects. The ﬁrst steps of this, which consist chieﬂy in the invention of new symbols of operation and a slight examination of their chief properties, I have endeavoured to give in the following pages.

I may now state what I hold to be the mission of Quaternions to Physics. I believe that Physics would advance with both more rapid and surer strides were Quaternions introduced to serious study to the almost total exclusion of Cartesian Geometry, except in an insigniﬁcant way as a particular case of the former. All the geometrical processes occurring in Physical theories and general Physical problems are much more graceful in their Quaternion than in their Cartesian garb. To illustrate what is here meant by “theory” and “general problem” let us take the case of Elasticity treated below. That by the methods advocated not only are the already well-known results of the general theory of Elasticity better proved, but more general results are obtained, will I think be acknowledged after a perusal of Section 7 to Section 11 below. That Quaternions are superior to Cartesian Geometry in considering the general problems of (1) an inﬁnite isotropic solid, (2) the torsion and bending of prisms and cylinders, (3) the general theory of wires, I have endeavoured to shew in Section 11–Section 13. But for particular problems such as the torsion problem for a cylinder of given shape, we require of course the various theories specially constructed for the solution of particular problems such as Fourier’s theories, complex variables, spherical harmonics, &c. It will thus be seen that I do not propose to banish these theories but merely Cartesian Geometry.

So mistaken are the common notions concerning the pretensions of advocates of Quaternions that I was asked by one well-known Mathematician whether Quaternions furnished methods for the solution of diﬀerential equations, as he asserted that this was all that remained for Mathematics in the domain of Physics! Quaternions can no more solve diﬀerential equations than Cartesian Geometry, but the solution of such equations can be performed as readily, in fact generally more so, in the Quaternion shape as in the Cartesian. But that the sole work of Physical Mathematics to-day is the solution of diﬀerential equations I beg to question. There are many and important Physical questions and extensions of Physical theories that have little or nothing to do with such solutions. As witness I may call attention to the new Physical work which occurs below.

If only on account of the extreme simplicity of Quaternion notation, large advances in the parts of Physics now indicated, are to be expected. Expressions which are far too cumbrous to be of much use in the Cartesian shape become so simple when translated into Quaternions, that they admit of easy interpretation and, what is perhaps of more importance, of easy manipulation. Compare for instance the particular case of equation (15 $m$ ) Section 9 below when $F = 0$ with the same thing as considered in Thomson and Tait’s Nat. Phil., App. C. The Quaternion equation is

\begin{array}{l} {ρ'}_{1} S \nabla_{1 Ψ} D w Δ = 0 . \end{array}

The Cartesian exact equivalent consists of Thomson and Tait’s equations (7), viz.

\begin{array}{l} \frac{d}{d x} & \{2 \frac{d w}{d A} (\frac{d α}{d x} + 1) + \frac{d w}{d b} \frac{d α}{d z} + \frac{d w}{d c} \frac{d α}{d y}\} \\ + \frac{d}{d y} \{2 \frac{d w}{d B} \frac{d α}{d y} + \frac{d w}{d a} \frac{d α}{d z} + \frac{d w}{d c} (\frac{d α}{d x} + 1)\} \\ + \frac{d}{d z} \{2 \frac{d w}{d C} \frac{d α}{d z} + \frac{d w}{d a} \frac{d α}{d y} + \frac{d w}{d b} (\frac{d α}{d x} + 1)\} = 0, \end{array}

and two similar equations.

Many of the equations indeed in the part of the essay where this occurs, although quite simple enough to be thoroughly useful in their present form, lead to much more complicated equations than those just given when translated into Cartesian notation.

It will thus be seen that there are two statements to make good:—(1) that Quaternions are in such a stage of development as already to justify the practically complete banishment of Cartesian Geometry from Physical questions of a general nature, and (2) that Quaternions will in Physics produce many new results that cannot be produced by the rival and older theory.

To establish completely the ﬁrst of these propositions it would be necessary to go over all the ground covered by Mathematical Physical Theories, by means of our present subject, and compare the proofs with the ordinary ones. This of course is impossible in an essay. It would require a treatise of no small dimensions. But the principle can be followed to a small extent. I have therefore taken three typical theories and applied Quaternions to most of the general propositions in them. The three subjects are those of Elastic Solids, with the thermodynamic considerations necessary, Electricity and Magnetism, and Hydrodynamics. It is impossible without greatly exceeding due limits of space to consider in addition, Conduction of Heat, Acoustics, Physical Optics, and the Kinetic Theory of Gases. With the exception of the ﬁrst of these subjects I do not profess even to have attempted hitherto the desired applications, but one would seem almost justiﬁed in arguing that, since Quaternions have been found so applicable to the subjects considered, they are very likely to prove useful to about the same extent in similar theories. Again, only in one of the subjects chosen, viz., Hydrodynamics, have I given the whole of the general theory which usually appears in text-books. For instance, in Electricity and Magnetism I have not considered Electric Conduction in three dimensions which, as Maxwell remarks, lends itself very readily to Quaternion treatment, nor Magnetic Induction, nor the Electro-Magnetic Theory of Light. Again, I have left out the consideration of Poynting’s theories of Electricity which are very beautifully treated by Quaternions, and I felt much tempted to introduce some considerations in connection with the Molecular Current theory of Magnetism. With similar reluctance I have been compelled to omit many applications in the Theory of Elastic Solids, but the already too large size of the essay admitted of no other course. Notwithstanding these omissions, I think that what I have done in this part will go far to bear out the truth of the ﬁrst proposition I have stated above.

But it is the second that I would especially lay stress upon. In the ﬁrst it is merely stated that Cartesian Geometry is an antiquated machine that ought to be thrown aside to make room for modern improvements. But the second asserts that the improved machinery will not only do the work of the old better, but will also do much work that the old is quite incapable of doing at all. Should this be satisfactorily established and should Physicists in that case still refuse to have anything to do with Quaternions, they would place themselves in the position of the traditional workmen who so strongly objected to the introduction of machinery to supplant manual labour.

But in a few months and synchronously with the work I have already described, to arrive at a large number of new results is too much to expect even from such a subject as that now under discussion. There are however some few such results to shew. I have endeavoured to advance each of the theories chosen in at least one direction. In the subject of Elastic Solids I have expressed the stress in terms of the strain in the most general case, i.e. where the strain is not small, where the ordinary assumption of no stress-couple is not made and where no assumption is made as to homogeneity, isotropy, &c. I have also obtained the equations of motion when there is given an external force and couple per unit volume of the unstrained solid. These two problems, as will be seen, are by no means identical. In Electrostatics I have considered the most general mechanical results ﬂowing from Maxwell’s theory, and their explanation by stress in the dielectric. These results are not known, as might be inferred from this mode of statement, for to solve the problem we require to know forty-two independent constants to express the properties of the dielectric at a given state of strain at each point. These are the six coeﬃcients of speciﬁc inductive capacity and their thirty-six diﬀerential coeﬃcients with regard to the six coordinates of pure strain. But, as far as I am aware, only such particular cases of this have already been considered as make the forty-two constants reduce at most to three. In Hydrodynamics I have endeavoured to deduce certain general phenomena which would be exhibited by vortex-atoms acting upon one another. This has been done by examination of an equation which has not, I believe, been hitherto given. The result of this part of the essay is to lead to a presumption against Sir William Thomson’s Vortex-Atom Theory and in favour of Hicks’s.

As one of the objects of this introduction is to give a bird’s-eye view of the merits of Quaternions as opposed to Cartesian Geometry, it will not be out of place to give side by side the Quaternion and the Cartesian forms of most of the new results I have been speaking about. It must be premised, as already hinted, that the usefulness of these results must be judged not by the Cartesian but by the Quaternion form.

Cartesian form of some of the results to follow

Elasticity

Let the point $(x, y, z)$ of an elastic solid be displaced to $(x', y', z')$ . The strain at any point that is caused may be supposed due to a pure strain followed by a rotation. In Section III. below, this pure strain is called $ψ$ . Let its coordinates be $e$ , $f$ , $g$ , $a ∕ 2$ , $b ∕ 2$ , $c ∕ 2$ ; i.e. if the vector $(ξ, η, ζ)$ becomes $(ξ', η', ζ')$ by means of the pure strain, then

\begin{matrix} ξ' = e ξ + \frac{1}{2} c η + \frac{1}{2} b ζ, \\ &c., &c. \end{matrix}

Thus when the strain is small $e$ , $f$ , $g$ reduce to Thomson and Tait’s $1 + e$ , $1 + f$ , $1 + g$ and $a$ , $b$ , $c$ are the same both in their case and the present one. Now let the coordinates of $Ψ$ , Section 9 below, be $E$ , $F$ , $G$ , $A ∕ 2$ , $B ∕ 2$ , $C ∕ 2$ . Equation (15), Section 9 below, viz.³

\begin{array}{l} Ψ ω = ψ^{2} ω = χ' χ ω = \nabla_{1} S {ρ'}_{1} {ρ'}_{2} S ω \nabla_{2}, \end{array}

gives in our present notation

\begin{array}{l} E & = e^{2} + c^{2} ∕ 4 + b^{2} ∕ 4 = {(d x' ∕ d x)}^{2} + {(d y' ∕ d x)}^{2} + {(d z' ∕ d x)}^{2}, \\ &c., &c. \\ A & = a (f + g) + b c ∕ 2 \\ = 2 {(d x' ∕ d y) (d x' ∕ d z) + (d y' ∕ d y) (d y' ∕ d z) + (d z' ∕ d y) (d z' ∕ d z)}, \\ &c., &c. \end{array}

which shew that the present $E$ , $F$ , $G$ , $A ∕ 2$ , $B ∕ 2$ , $C ∕ 2$ are the $A$ , $B$ , $C$ , $a$ , $b$ , $c$ of Thomson and Tait’s Nat. Phil., App. C.

Let us put

\begin{array}{l} J (\begin{matrix} x' & y' & z' \\ x & y & z \end{matrix}) & = J \\ J (\begin{matrix} y' & z' \\ y & z \end{matrix}) & = J_{11}, &c., &c., \\ J (\begin{matrix} z' & x' \\ y & z \end{matrix}) & = J_{12}, J (\begin{matrix} y' & z' \\ z & x \end{matrix}) = J_{21}, &c., &c., &c., &c. \end{array}

I have shown in Section 8 below that the stress-couple is quite independent of the strain. Thus we may consider the stress to consist of two parts—an ordinary stress $P Q R S T U$ as in Thomson and Tait’s Nat. Phil. and a stress which causes a couple per unit volume $L' M' N'$ . The former only of these will depend on strain. The result of the two will be to cause a force (as indeed can be seen from the expressions in Section 7 below) per unit area on the $x$ -interface $P$ , $U + N' ∕ 2$ , $T - M' ∕ 2$ , and so for the other interfaces. If $L$ , $M$ , $N$ be the external couple per unit volume of the unstrained solid we shall have

\begin{array}{l} L' = - L ∕ J, M' = - M ∕ J, N' = - N ∕ J, \end{array}

for the external couple and the stress-couple are always equal and opposite. Thus the force on the $x$ -interface becomes

\begin{array}{l} P, U - N ∕ 2 J, T + M ∕ 2 J \end{array}

and similarly for the other interfaces.

To express the part of the stress ( $P$ &c.) which depends on the strain in terms of that strain, consider $w$ the potential energy per unit volume of the unstrained solid as a function of $E$ &c. In the general thermodynamic case $w$ may be deﬁned by saying that

\begin{array}{l} w & \times (the element of volume) \\ = (the intrinsic energy of the element) \\ - (the entropy of the element \times its absolute temperature \times Joule’s coeﬃcient) . \end{array}

Of course $w$ may be, and indeed is in Section 8, Section 8 below, regarded as a function of $e$ &c.

The equation for stress is (15 $b$ ) Section 9 below, viz.,

\begin{array}{l} J \bar{ϕ} ω = 2 χ Ψ D w χ' ω = 2 ρ' S ρ' ω S \nabla_{1 Ψ} D w \nabla_{2} . \end{array}

The second of the expressions is in terms of the strain and the third in terms of the displacement and its derivatives. In our present notation this last is

\begin{array}{l} \frac{J P}{2} & = {(\frac{d x'}{d x})}^{2} \frac{d w}{d E} + {(\frac{d x'}{d y})}^{2} \frac{d w}{d F} + {(\frac{d x'}{d z})}^{2} \frac{d w}{d G} \\ b e g i n a l i g n * 1 e x] & + 2 \frac{d x'}{d y} \frac{d x'}{d z} \frac{d w}{d A} + 2 \frac{d x'}{d z} \frac{d x'}{d x} \frac{d w}{d B} + 2 \frac{d x'}{d x} \frac{d x'}{d y} \frac{d w}{d C}, \\ &c., &c. \\ \frac{J S}{2} & = \frac{d y'}{d x} \frac{d z'}{d x} \frac{d w}{d E} + \frac{d y'}{d y} \frac{d z'}{d y} \frac{d w}{d F} + \frac{d y'}{d z} \frac{d z'}{d z} \frac{d w}{d G} \\ b e g i n a l i g n * 1 e x] & + (\frac{d y'}{d y} \frac{d z'}{d z} + \frac{d y'}{d z} \frac{d z'}{d y}) \frac{d w}{d A} + (\frac{d y'}{d z} \frac{d z'}{d x} + \frac{d y'}{d x} \frac{d z'}{d z}) \frac{d w}{d B} \\ b e g i n a l i g n * 1 e x] & + (\frac{d y'}{d x} \frac{d z'}{d y} + \frac{d y'}{d y} \frac{d z'}{d x}) \frac{d w}{d C}, \\ &c., &c. \end{array}

In Section 8 I also obtain this part of the stress explicitly in terms of $e$ , $f$ , $g$ , $a$ , $b$ , $c$ , of $w$ as a function of these quantities and of the axis and amount of rotation. But these results are so very complicated in their Cartesian shape that it is quite useless to give them.

To put down the equations of motion let $X_{x}$ , $Y_{x}$ , $Z_{x}$ be the force due to stress on what before strain was unit area perpendicular to the axis of $x$ . Similarly for $X_{y}$ , &c. Next suppose that $X$ , $Y$ , $Z$ is the external force per unit volume of the unstrained solid and let $D$ be the original density of the solid. Then the equation of motion (15 $n$ ) Section 9 below, viz.

\begin{array}{l} D \ddot{ρ}' = F + τ Δ, \end{array}

gives in our present notation

\begin{array}{l} X + d X_{x} ∕ d x + d X_{y} ∕ d y + d X_{z} ∕ d z = \ddot{x}' D, &c., &c. \end{array}

It remains to express $X_{x}$ &c. in terms of the displacement and $L M N$ . This is done in equation (15 $l$ ) Section 9 below, viz⁴.

\begin{array}{l} τ ω = - 2 ρ' S \nabla_{1 Ψ} D w ω + 3 V M V ρ' ρ' S ω \nabla_{1} \nabla_{2} ∕ 2 S \nabla_{1} \nabla_{2} \nabla_{3} S ρ' ρ' ρ' . \end{array}

In our present notation this consists of the following nine equations:

\begin{array}{l} X_{x} & = 2 (\frac{d w}{d E} \frac{d x'}{d x} + \frac{d w}{d C} \frac{d x'}{d y} + \frac{d w}{d B} \frac{d x'}{d z}) + \frac{J_{12} N - J_{13} M}{2 J}, \\ Y_{x} & = 2 (\frac{d w}{d E} \frac{d y'}{d x} + \frac{d w}{d C} \frac{d y'}{d y} + \frac{d w}{d B} \frac{d y'}{d z}) + \frac{J_{13} L - J_{11} N}{2 J}, \\ Z_{x} & = 2 (\frac{d w}{d E} \frac{d z'}{d x} + \frac{d w}{d C} \frac{d z'}{d y} + \frac{d w}{d B} \frac{d z'}{d z}) + \frac{J_{11} M - J_{12} L}{2 J}, \end{array}

and six similar equations.

We thus see that in the case where $L M N$ are zero, our present $X_{x}$ , $X_{y}$ , $X_{z}$ are the $P Q R$ of Thomson and Tait’s Nat. Phil. App. C (d), and therefore equations (7) of that article agree with our equations of motion when we put both the external force and the acceleration zero.

These are some of the new results in Elasticity, but, as I have hinted, there are others in Section 8, Section 8 which it would be waste of time to give in their Cartesian form.

Electricity

In Section IV. below I have considered, as already stated, the most general mechanical results ﬂowing from Maxwell’s theory of Electrostatics. I have shewn that here, as in the particular cases considered by others, the forces, whether per unit volume or per unit surface, can be explained by a stress in the dielectric. It is easiest to describe these forces by means of the stress.

Let the coordinates of the stress be $P Q R S T U$ . Then $F_{1} F_{2} F_{3}$ the mechanical force, due to the ﬁeld per unit volume, exerted upon the dielectric where there is no discontinuity in the stress, is given by

\begin{array}{l} F_{1} = d P ∕ d x + d U ∕ d y + d T ∕ d z, &c., &c. \end{array}

and $(l, m, n)$ being the direction cosines of the normal to any surface, pointing away from the region considered

\begin{array}{l} F' = - {[l P + m U + n T]}_{a} - {[]}_{b}, &c., &c., \end{array}

where $a$ , $b$ indicate the two sides of the surface and $F_{1}'$ , $F_{2}'$ , $F_{3}'$ is the force due to the ﬁeld per unit surface.

It remains to ﬁnd $P$ &c. Let $X$ , $Y$ , $Z$ be the electro-motive force, $α$ , $β$ , $γ$ the displacement, $w$ the potential energy per unit volume and $K_{x x}$ , $K_{y y}$ , $K_{z z}$ , $K_{y z}$ , $K_{z x}$ , $K_{x y}$ the coeﬃcients of speciﬁc inductive capacity. Let $1 + e$ , $1 + f$ , $1 + g$ , $a ∕ 2$ , $b ∕ 2$ , $c ∕ 2$ denote the pure part of the strain of the medium. The $K$ ’s will then be functions of $e$ &c. and we must suppose these functions known, or at any rate we must assume the knowledge of both the values of the $K$ ’s and their diﬀerential coeﬃcients at the particular state of strain in which the medium is when under consideration. The relations between the above quantities are

\begin{matrix} 4 π α = K_{x x} X + K_{x y} Y + K_{z x} Z, &c., &c. \\ \begin{aligned} w & = (X α + Y β + Z γ) ∕ 2 \\ = (K_{x x} X^{2} + K_{y y} Y^{2} + K_{z z} Z^{2} + 2 K_{y z} Y Z + 2 K_{z x} Z X + 2 K_{x y} X Y) ∕ 8 π . \end{aligned} \end{matrix}

It is the second of these expressions for $w$ which is assumed below, and the diﬀerentiations of course refer only to the $K$ ’s. The equation expressing $P$ &c. in terms of the ﬁeld is (21) Section IV below, viz.

\begin{array}{l} ϕ ω = - \frac{1}{2} V D ω E - Ψ D w ω, \end{array}

which in our present notation gives the following six equations

\begin{array}{l} P & = - \frac{1}{2} (- α X + β Y + γ Z) - d w ∕ d e, &c., &c., \\ S & = \frac{1}{2} (β Z + γ Y) - d w ∕ d a, &c., &c. \end{array}

I have shown in Section 14–Section 14 below that these results agree with particular results obtained by others.

Hydrodynamics

The new work in this subject is given in Section VI.—“The Vortex-Atom Theory.” It is quite unnecessary to translate the various expressions there used into the Cartesian form. I give here only the principal equation in its two chief forms, equation (9) Section 25 and equation (11) Section 25, viz.

\begin{array}{l} P + v - σ^{2} ∕ 2 + {(4 π)}^{- 1} ∭ (S σ τ \nabla u + u \partial m ∕ \partial t) d 𝔰 = H, \\ P + v - σ^{2} ∕ 2 + {(4 π)}^{- 1} ∭ {d 𝔰 S \nabla u (V σ τ - m σ) + u d (m d 𝔰) ∕ d t} = H . \end{array}

In Cartesian notation these are

\begin{array}{l} \int d p ∕ ρ + V + q^{2} ∕ 2 \\ - {(4 π)}^{- 1} ∭ {2 [(x' - x) (w η - v ζ) + \dots + \dots] ∕ r^{3} \\ + (\partial c ∕ \partial t) ∕ r} d x' d y' d z' = H . \\ \int d p ∕ ρ + V + q^{2} ∕ 2 \\ - {(4 π)}^{- 1} ∭ {(x' - x) [2 (w η - v ζ) - c u] + \dots + \dots} ∕ r^{3} . d x' d y' d z' \\ - {(4 π)}^{- 1} ∭ {d (c d x' d y' d z') ∕ d t} ∕ r = H . \end{array}

The ﬂuid here considered is one whose motion is continuous from point to point and which extends to inﬁnity. The volume integral extends throughout space. The notation is as usual. It is only necessary to say that $H$ is a function of the time only, $r$ is the distance between the points $x', y', z'$ and $x, y, z$ ;

\begin{array}{l} c = d u ∕ d x + d v ∕ d y + d w ∕ d z; \end{array}

$d ∕ d t$ is put for diﬀerentiation which follows a particle of the ﬂuid, and $\partial ∕ \partial t$ for that which refers to a ﬁxed point.

The explanation of the unusual length of this essay, which I feel is called for, is contained in the foregoing description of its objects. If the objects be justiﬁable, so must also be the length which is a necessary outcome of those objects.

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Quaternions as a practical instrument of physical research IIntroduction

Quaternions as a practical instrument of physical research

Cartesian form of some of the results to follow

Electricity

Hydrodynamics

Quaternions as a practical instrument of physical research

I
Introduction