Notes

¹[Note added, 1892. It would be better to head this section “Elastic bodies” since except when the strains are assumed small the equations are equally true of solids and fluids. I may say here that I have proved in the Proc. R. S. E. 1890–91, pp. 106 et seq., most of the general propositions of this section somewhat more neatly though the processes are essentially the same as here.]

²Elect. and Mag. Vol. I. § 10.

³This result is one of Tait’s (Quaternions § 365 where he has $\varphi \prime \varphi ={\overline{\varpi }}^2$). It is given here for completeness.

⁴The second term on the right contains in full the nine terms corresponding to . Quaternion notation is therefore here, as in nearly all cases which occur in Physics, considerably more compact even than the notations of determinants or Jacobians.

⁵These meanings for ${\nabla }_1,{\nabla }_2\dots \Delta$ I used in a paper on “Some General Theorems in Quaternion Integration,” in the Mess. of Math. Vol. xiv. (1884), p. 26. The investigations there given are for the most part incorporated below. [Note added, 1892, see preface as to the alteration of $\nabla \prime$ into $\Delta$.]

⁶I have used an inverted $D$ to indicate the analogy to Hamilton’s inverted $\Delta$.

⁷[Note added, 1892. For practice it is convenient to remember this in words:—A term in which $\zeta$ and $\varphi \zeta$ occur is unaltered in value by changing them into $\varphi \prime \zeta$ and $\zeta$ respectively.]

⁸[Note added, 1892. The cubic may be obtained in a more useful form from the equation $\omega S{\zeta }_1{\zeta }_2{\zeta }_{3}S\varphi {\zeta }_1\varphi {\zeta }_2\varphi {\zeta }_3=-3V{\zeta }_1{\zeta }_{2}S\varphi \omega \varphi {\zeta }_1\varphi {\zeta }_2$ thus $$\begin{array}{llll}\hfill V{\zeta }_1{\zeta }_{2}S\varphi \omega \varphi {\zeta }_1\varphi {\zeta }_2& =\varphi \omega S\blacksquare \varphi {\zeta }_1\varphi {\zeta }_{2}V{\zeta }_1{\zeta }_2-\varphi {\zeta }_{1}S\blacksquare \varphi \omega \varphi {\zeta }_{2}V{\zeta }_1{\zeta }_2+\varphi {\zeta }_{2}S\blacksquare \varphi \omega \varphi {\zeta }_{1}V{\zeta }_1{\zeta }_2& \hfill & \\ \hfill & =\varphi \omega S\blacksquare \varphi {\zeta }_1\varphi {\zeta }_{2}V{\zeta }_1{\zeta }_2-2\varphi {\zeta }_{1}S\blacksquare \varphi \omega \varphi {\zeta }_{2}V{\zeta }_1{\zeta }_2.& \hfill & \end{array}$$

⁹[Note added, 1892. The following slightly more general statement is a practically much more convenient form of enunciation: if $S\chi \zeta \varphi \zeta =S\chi \zeta \psi \zeta$, where $\chi$ is a perfectly arbitrary self-conjugate and $\varphi$ and $\psi$ are not necessarily self-conjugate then $\overline{\varphi }=\overline{\psi }$].

¹⁰These two propositions are generalisations of what Tait and Hicks have from time to time proved. They were first given in the present form by me in the article already referred to in Section 1 above. In that paper the necessary references are given.

¹¹[Note added, 1892. Let me disarm criticism by confessing that what follows concerning $V\nabla \sigma$ is nonsense.]

¹²[Note added, 1892. It would be better to head this section “Elastic bodies” since except when the strains are assumed small the equations are equally true of solids and fluids. I may say here that I have proved in the Proc. R. S. E. 1890–91, pp. 106 et seq., most of the general propositions of this section somewhat more neatly though the processes are essentially the same as here.]

¹³[Note added, 1892. It would be better to head this section “Elastic bodies” since except when the strains are assumed small the equations are equally true of solids and fluids. I may say here that I have proved in the Proc. R. S. E. 1890–91, pp. 106 et seq., most of the general propositions of this section somewhat more neatly though the processes are essentially the same as here.]

¹⁴Observe that we do not make this assumption. We really shew that it is true.

¹⁵By putting in this equation, equation (11) can be deduced but as this method has already been applied in Section 2 I give the one in the text to shew the variety of Quaternion methods. [Note added, 1892. If we use the theorem in the foot-note of Section 2, equation (11) follows at once.]

¹⁶See Section 21 below.

¹⁷[Note added, 1892. In the original essay there was a slip here which I have corrected. It was caused by assuming that $\psi$ instead of $\psi -1$ was small for small strains. In the original I said “where $y$ is a constant and $x$ is a multiple of $a$.”]

¹⁸[By assuming from the second law that the work done by the element in the cycle, i.e. the sum of the works done by it during the first and third steps is $J\delta t∕t$ multiplied by the heat absorbed by the element in the third step. Note added, 1893.]

¹⁹[Note added, 1892. Better thus:—by Section 2 above,

therefore by Section 2,
for regarded as a function of $\omega$ is clearly self conjugate.]

²⁰[Note added, 1892. For a neater quaternion treatment of this problem see Phil. Mag. June, 1892, p. 493.]

²¹[Note added, 1892. This is not legitimate since it makes $F=\infty$ for $r=0$. The reasoning is rectified in the Phil. Mag. paper just referred to by putting $F=0$ from $r=0$ to $r=b$ and $Fr=r^{-a}\lambda$ from $r=b$ to $r=\infty$.]

²²[Note added, 1892. This is contrary to the usual convention.]

²³[Note added, 1892. In the Phil. Mag. June, 1892, p. 488, there is a mistake in the equation just preceding equation 31 and there are two mistakes in equation 31. In the first of these should be . In equation (31) all the 2’s should be dropped.]

²⁴This could be deduced from Tait’s Quaternions, § 356, equation (2). His $𝜖$ is our $q\omega q^{-1}$ and his dots our dashes.

²⁵Prof. J. J. Thomson in his paper on Electrical Theories, B.A. Reports, 1885, p. 125, does not credit Maxwell with such a definite and circumscribed theory as that described in the text, and he is thereby led to find fault with Maxwell’s term “Displacement” and points out that there is an assumption made with reference to the connection between the true current and this polarisation (displacement). He says moreover, “It is rather difficult to see what is meant in Maxwell’s Theory by the phrase ‘Quantity of Electricity.’ ” None of these remarks are called for if the view I take of Maxwell’s theory be correct, and these grounds alone I consider sufficient for taking that view. The paper of Thomson’s here mentioned I shall frequently have to refer to. [Note added, 1892. In the text I have given much too rigid a form to Maxwell’s theory. What I have called his theory I ought rather to have called his analogy. Still I think the present foot-note is in the main just. In my opinion it is no more and no less difficult to see what is meant in Maxwell’s Theory by “Quantity of Electricity” than by “displacement” since the two are connected by perfectly definite equations. Of course it is wrong to define “displacement” as “displacement of quantity of electricity” and then to define “quantity of electricity” in terms of “displacement” but Maxwell does not seem to me even tacitly to do this. Rather he says—the dielectric is polarised; this polarisation can be represented by a vector $D$; electrical quantity can be expressed in terms of $D$; the mathematical connections between electrical quantity and $D$ are the same as those between quantity of matter in a space and the displacement out of that space made by other matter to make room for the given matter; we will impress this useful analogy firmly on our minds by calling $D$ the displacement. But I have expressed my present views on the meaning of Maxwell’s theory much more fully in Phil. Trans. 1892, p. 685.]

²⁶This frequently recurring cumbrous mode of description must be tolerated unless a single word can be invented for “a linear vector function of a vector.” Might I suggest the term “Hamiltonian?” Thus we should say that the displacement is a Hamiltonian of the electro-motive force, the Hamiltonian at any point being a function of the state of the medium.

²⁷We see from this that $D=E\nabla w$ or $E=D\nabla w$ according as $w$ is looked upon as a function of $E$ or $D$.

²⁸This is for the general case following the example of Helmholtz in the particular case when $K$ reduces to a single scalar. See Wiss. Abh. vol. i. equation (2$d$), p. 805. The method adopted in the following investigation is also similar to his.

²⁹As far as I am aware nobody has hitherto attempted to find the electrical forces much less the stress except in the case when $D$ is parallel to $E$ i.e. the dielectric is electrically isotropic when unstrained. The particular results contained in Section 14 below have been obtained by Korteweg, Lorberg and Kirchhoff as is stated in Prof. J. J. Thomson’s paper (p. 155) referred to in Section IV.

³⁰This is the same result as Helmholtz’s on the same assumption Wiss. Abh. i. p. 798.

³¹For references to former proofs of this see foot-note to Section IV above.

³²[Note added, 1892. More generally and better .]

³³See Section 1 above for the convention with respect to the relation between the positive side of a surface and the positive direction round its boundary. Hitherto we have had no reason for choosing either the right-handed or the left-handed screw as the type of positive and negative rotation. But to make the statement in the text correct we must take the former.

³⁴I do not defend the legitimacy of this assumption. It seems to me bold to assume that a magnet possesses any such thing as potential energy in a field which has no potential. If we assume $H$ and its derivatives to be continuous throughout our typical element $d\mathfrak{𝔰}$ of volume containing a great number of molecules (both material and magnetic) the force on a magnetic molecule $\mu$ consisting of two poles is $-S\mu \nabla \blacksquare H$ and the force per unit volume $-SI\nabla \blacksquare H$, which is only identical with the expression $-{\nabla }_{1}SI{H}_1$ obtained below when $H$ has a potential. With this expression a stress cannot be found that produces the force. If, however, $H$ and its derivatives be not assumed continuous in this manner the force on the magnet $\mu$ is quite indeterminate whether the magnetic pole or the molecular current view of magnetism be taken, unless it be specified in what way the poles and currents are distributed in the element of volume.

³⁵[Note added, 1892. Prof. Tait’s name ought to be added to Prof. Hicks’s.]

³⁶[Note added, 1892. I am aware that this is contrary to the usual English custom, but that custom—of interchanging the meanings of $d∕dt$ and $\partial ∕\partial t$ as given in the text—seems to me out of harmony with the meaning attached to $\partial$ in other branches of Mathematics. At any rate I have respectable fellow-sinners, e.g. Kirchhoff in his Mechanik, zweite Vorlesung, et seq.]

³⁷This is quoted as a known result because it occurs generally in the subject of Rigid Dynamics. No Quaternion proof, however as far as I am aware, has been given. We therefore give one here. What is meant by rotating axes may be thus explained.—Instead of choosing as our coordinates the vectors $\alpha \prime$, $\beta \prime \dots$ which occur in any problem, we take others $\alpha$, $\beta \dots$ such that $\alpha \prime =q\alpha q^{-1}$, $\beta \prime =q\beta q^{-1}\dots$ may be called the integral rotation of the axes. Thus if we say that the vector angular velocity of these axes themselves is $\omega$ we mean that the real angular velocity is $q\omega q^{-1}$, so that, as can be seen by putting in equation ($a$) below $\alpha =\text{ const.}$, or as in equation (58) Section 13 above, $\omega =2V{q}^{-1}\dot{q}$. (This maybe established also by Tait’s Quaternions, § 356, equation (2), from which $q\omega q^{-1}=2V\dot{q}{q}^{-1}$ or $\omega =2V{q}^{-1}\dot{q}$). Again, when we say that the rate of increase of $\alpha$ in space is $\tau$, we mean that $\dot{\alpha }\prime =q\tau q^{-1}$ or , or

$$\begin{array}{llllllllllll}\hfill & & \hfill \tau & =\dot{\alpha }+2VV{q}^{-1}\dot{q}\blacksquare \alpha ,& \hfill & & \hfill & & \hfill & & \hfill & \text{(}a\text{)}\\ \hfill & \text{ or}& \hfill \tau & =\dot{\alpha }+V\omega \alpha .& \hfill & & \hfill & & \hfill & & \hfill & \text{(}b\text{)}\end{array}$$

This could have been proved with fewer symbols and more explanation, but the above seems to me the most characteristic Quaternion proof. We might have started with not quite so general an explanation of reference to rotating axes and so refrained from introducing the integral rotation, and therefore also $q$.

³⁸[Note added, 1892. At the time of writing the essay I did not notice that these equations are a particular case of the general equation for an elastic body already established (see equations (15$m$) Section 9 and (15$n$) Section 9)

$$\begin{array}{lll}\hfill \left.D\ddot{\rho }\prime =F-2{\rho \prime }_{1}S{\nabla }_1\Psi Dw\Delta .\right]& & \hfill \end{array}$$

³⁹[Note added, 1892. When first giving this in the Mess. of Math. 1884 and when again putting it in the present essay, I was unaware of Prof. Tait’s paper in Proc. R. S. E. 1869–70, p. 143, where in the present notation he has for an incompressible fluid, (1) $d∕dt=\partial ∕\partial t-S\sigma \nabla$ [given in Cartesian form], (2) $\dot{\sigma }=-\nabla v-\nabla p∕D$, (3) $S\nabla \sigma =0$, (4) , (5) $V\nabla \dot{\sigma }=0$, (6) $d\nabla \sigma ∕dt=-S\nabla \sigma \nabla \blacksquare \sigma$. Perhaps I have not interpreted Prof. Tait’s notation which is but briefly described correctly, but (4) should apparently be , and so agree with (5) and (6). It will be seen that the whole of §§ 512–13 of Tait’s Quaternions, 3rd ed., is contained in this essay. I cannot at present recall whether this is owing wholly or in part to my being indebted to other old papers of Prof. Tait, or whether in writing his 3rd ed. he has arrived independently at the same treatment, but am inclined to the latter belief.]

⁴⁰To insure that $\varphi$ is single valued and so, that the part of the surface integral of equation (29) due to “barriers” is zero.

⁴¹For let be this general function $A$, $B$, and $C$ being scalar functions each linear in each of its constituents, and let

⁴²Kelland and Tait’s Quaternions, chap. x. equation ($n$). We have already used a particular case of this in Section 8, above.

⁴³By “bubbles” of course I do not mean spaces occupied by another kind of fluid of smaller density but actual vacua. Thus a bubble may start into existence where none previously existed, or again a bubble may completely disappear.

⁴⁴There is one important difference to be noticed between this and Hicks’s explanation of gravitation. His depends on the synchronous pulsations of distant vortices. I do not wish to imply that I do not believe in the existence of such synchronous pulsations, but by the above we see that gravitation can probably be explained independently of them. As a matter of fact such synchronous pulsations probably actually occur on account of the variation of $H$ with the time.

⁴⁵[Note added, 1892. Because the density of attracting magnetic matter of the analogue $=-S\nabla H∕4\pi =-m∕4\pi$.]

⁴⁶[Note added, 1892. This should be , and therefore equation (29) should be

This does not affect our present problem because $m=0$ in our case.]

⁴⁷One way out of many of proving this is as follows. Divide the bubble up into a number of infinitely near sections by planes perpendicular to the unit vector $\alpha$. For any one section $\int \; <!--nolimits-->Sd\rho \sigma =0$. Consider $d\Sigma$ to be the element of the surface cut off by the following four planes, (1) the plane of section considered, (2) the consecutive plane of section, (3) the two planes perpendicular to $d\rho$, and through the extremities of the element $d\rho$. Thus if $x$ be the distance between the two sections, $d\rho =x^{-1}V\alpha d\Sigma$, whence

$$\begin{array}{lll}\hfill \iint S\alpha d\Sigma \sigma =0,& & \hfill \end{array}$$

for the surface of the bubble between the two sections. But adding, we may suppose this integral to extend over the whole bubble. Thus $S\alpha \iint Vd\Sigma \sigma =0$ for the whole bubble; therefore $\alpha$ being a quite arbitrary unit vector we have for the whole bubble $\iint Vd\Sigma \sigma =0$.

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