VI The Vortex-Atom Theory

VI
The Vortex-Atom Theory

85. If Quaternions can give valuable hints or indicate a promising method of dealing with the highly interesting mathematical theory of Vortex-Atoms, I think this alone ought to be suﬃcient defence of its claims to be within the range of practical methods of investigation.

In what follows I think I may be said to have indicated a hopeful path to follow in order to test to some extent the soundness of this theory.

22. Statement of Sir Wm. Thomson’s and Prof. Hicks’s theories

86. Sir Wm. Thomson’s theory is so well-known that it is not necessary to state it in detail. Matter is some diﬀerentiation of space which can vary its position carrying with it so to speak certain phenomena, some of which admit of deﬁnite quantitative measurement. Perhaps the most important of these phenomena is what is called mass. Now, says in eﬀect Sir Wm. Thomson, if we suppose all space ﬁlled with an incompressible perfect ﬂuid the vortices in it are just such diﬀerentiations of space. They also carry about with them deﬁnite quantitative phenomena. Can we prove that these hypothetical vortices would act upon one another as do the atoms of matter, the laws of whose action are contained in the various Sciences, e.g. Physics, Chemistry and Physiology? The problem in its ﬁrst stages at any rate is a mathematical one, but during the many years it has been before the mathematical world very little progress has been made with it.

Hicks’s extension of this theory is perhaps not so well known, but it seems to me quite as interesting and more likely to tally with the known phenomena of matter. He enunciates his theory in the Proceedings of the Cambridge Philosophical Society, vol. iii. p. 276. It diﬀers from Thomson’s simply in assuming that the ﬂuid does not quite ﬁll space—that there are in it bubbles⁴³. These bubbles will ﬁnd their way to where the pressure is least, i.e. speaking generally to the centre of some at least of the vortices. Thus we have another source of diﬀerentiation of space and general considerations seem to point to these diﬀerentiations being the true atoms, though of course “atom” is no longer a descriptive term.

23. General considerations concerning these theories

87. In Maxwell’s article Atom in the Encyc. Brit. p. 45, he says—“One of the ﬁrst if not the very ﬁrst desideratum in a complete theory of matter is to explain ﬁrst mass and second gravitation…… In Thomson’s theory the mass of bodies requires explanation. We have to explain the inertia of what is only a mode of motion and inertia is a property of matter, not of modes of motion. It is true that a vortex ring at any given instant has a deﬁnite momentum and a deﬁnite energy, but to shew that bodies built up of vortex rings would have such momentum and energy as we know them to have is in the present state of the theory a very diﬃcult task.

“It may seem hard to say of an infant theory that it is bound to explain gravitation.”

Now as Hicks tells us what induced him to give his theory was the promise it gave of explaining gravitation. But I believe nowhere does he point out the still more important result that probably on his theory we can explain inertia.

The statement of the principal property of inertia put scientiﬁcally is that the motion of the centre of gravity of any two bodies approximates more and more nearly to uniform velocity in a straight line, the more nearly they are isolated from external inﬂuence. But this property is probably true of Hicks’s bubbles. The centre of gravity of any portion of the ﬂuid containing certain bubbles $(1)$ , $(2) \dots (n)$ will if approximately isolated from all the rest of the ﬂuid move approximately in a straight line, but this amounts to saying that the centre of volume of the $n$ bubbles will also move uniformly in a straight line if the centre of the whole volume (bubbles and ﬂuid) considered, move similarly. These conditions are not evidently true, but I think they are probably so when we consider groups of large numbers of bubbles.

Whether this theory explains gravitation is one of the principal questions to be considered in the following ﬁrst trial.

24. Description of the method here adopted

88. Finding it practically impossible to consider the real problem of a number of bubbles in the liquid, I consider the ﬂuid continuous and not incompressible. Let us assume that

\begin{array}{l} D = tanh (p ∕ c), & (1) \end{array}

where $c$ is some very small constant which in the limit $= 0$ . For ordinary values of $p$ , $D very nearly = 1$ . It is only when $p$ becomes comparable with $c$ that $D$ varies. When $p = 0$ , $D = 0$ . Also

\begin{array}{l} P = \int (d p ∕ D) = c log sinh (p ∕ c) . & (2) \end{array}

For ordinary values of $p$ then, $P = p$ but when $p$ becomes comparable with $c$ , $P$ varies and when $p = 0$ , $P = - \infty$ . If we assume then for the greater part of the ﬂuid that $p$ is large compared with the small quantity $c$ we see that the ﬂuid will have almost exactly the same properties as a liquid containing bubbles. We may now apply the equations of motion of Section 15, Section 15.

Now we know the velocity at any point of an inﬁnite ﬂuid in terms of the spins and convergences at all points. From this we may deduce the acceleration in terms of the spins, convergences and time-ﬂuxes. It so happens that by the equations in Section 17 we can get rid of the time-ﬂuxes of the spins. This greatly simpliﬁes the further discussion of the problem. The equation that we thus deduce forms the starting point of our investigation, for the phenomena of gravitation, electro-magnetism, stress, &c., are exhibited and measured by the acceleration in bodies due to their relative positions.

The equation we obtain at once gives a generalisation of the integral, equation (12) Section 15 above.

At the end of this section I give an equation which is rather complicated but which promises to enable us to deal more rigorously with our problem than I profess to have done below.

We proceed to the investigation just indicated.

25. Acceleration in terms of the convergences, their time-ﬂuxes, and the spins

89. Let us put for the convergence and twice the spin $m$ and $τ$ respectively, so that

\begin{array}{l} S \nabla σ & = m, & (3) \\ V \nabla σ & = τ . & (4) \end{array}

We have seen that in equation (50) Section 20 we may neglect the surface integral. Thus

\begin{array}{l} 4 π σ & = \nabla ∭ u (m + τ) d 𝔰, & (5) \\ ∴ 4 π \partial σ ∕ \partial t & = \nabla ∭ u \partial (m + τ) ∕ \partial t · d 𝔰 . \\ By Section 17 & \dot{τ} & = τ S \nabla σ - S τ \nabla · σ, \end{array}

but by equation (1) Section 15

\begin{array}{l} \partial τ ∕ \partial t & = \dot{τ} + S σ \nabla · τ, \\ ∴ \partial τ ∕ \partial t & = S σ \nabla · τ + τ S \nabla σ - S τ \nabla · σ, \\ or because S \nabla τ = 0 \\ \partial τ ∕ \partial t & = τ S Δ σ - σ S Δ τ = V \nabla V σ τ & (6) \\ ∴ 4 π \partial σ ∕ \partial t & = \nabla ∭ u (V \nabla V σ τ + \partial m ∕ \partial t) d 𝔰 . & (7) \end{array}

This equation is not in a convenient form for our purpose, for $τ$ and $m$ may be and most probably will be discontinuous, so that it is advisable to get rid of their derivatives. Moreover it is advisable to allow the time-ﬂux of $m$ to appear only under the form $d (m d 𝔰) ∕ d t$ because in case of pulsations the time integral of this expression will be zero, whereas we can predicate no such thing of $(\partial m ∕ \partial t) d 𝔰$ , or indeed of $ṁ d 𝔰$ . Let us ﬁrst then consider the ﬁrst term in equation (7), viz. $\nabla ∭ u V \nabla V σ τ d 𝔰$ . Using equation (9) Section 4, and neglecting the surface integral at inﬁnity, this becomes

\begin{array}{l} - \nabla ∭ V \nabla u V σ τ d 𝔰 = \nabla V \nabla ∭ u V σ τ d 𝔰 & [Section 5] \\ = \nabla^{2} ∭ u V σ τ d 𝔰 - \nabla S \nabla ∭ u V σ τ d 𝔰 \\ = 4 π V σ τ + \nabla ∭ S σ τ \nabla u d 𝔰, \end{array}

by equation (19) Section 5 above.

Remembering now (Section 15 above), that

\begin{array}{l} \dot{σ} = \partial σ ∕ \partial t - V σ τ - \nabla (σ^{2}) ∕ 2, \end{array}

we see by equation (7) that

\begin{array}{l} 4 π \dot{σ} = - 2 π \nabla · σ^{2} + \nabla ∭ S σ τ \nabla u d 𝔰 + \nabla ∭ u (\partial m ∕ \partial t) d 𝔰, & (8) \end{array}

whence by equation (9) Section 15 above

\begin{array}{l} P + v - σ^{2} ∕ 2 + {(4 π)}^{- 1} ∭ (S σ τ \nabla u + u \partial m ∕ \partial t) d 𝔰 = H, & (9) \end{array}

where $H$ is a function of the time only. This is a generalisation of equation (12) Section 15, for assuming that $τ = 0$ we know by Section 20 that $\partial ϕ ∕ \partial t = {(4 π)}^{- 1} ∭ u (\partial m ∕ \partial t) d 𝔰$ .

90. This integral equation may be put into several diﬀerent forms by means of equation (9) Section 4 above. The form that is useful to us is the one in which instead of $\partial m ∕ \partial t$ we have $d (m d 𝔰) ∕ d t$ as we have already seen. Now $d (d 𝔰) ∕ d t = - m d 𝔰$ . Therefore

\begin{array}{l} d (m d 𝔰) ∕ d t & = (ṁ - m^{2}) d 𝔰 \\ = (\partial m ∕ \partial t - S σ \nabla m - m^{2}) d 𝔰 . \end{array}

Substituting from this for $\partial m ∕ \partial t$ , and then transforming by equation (9) Section 4 so as to get rid of the space variations of $m$ involved in $S σ \nabla m$ we get

\begin{array}{l} ∭ u (\partial m ∕ \partial t) d 𝔰 & = ∭ u d (m d 𝔰) ∕ d t + ∭ (u m^{2} - m S σ \nabla u - u m^{2}) d 𝔰 \\ = ∭ u d (m d 𝔰) ∕ d t - ∭ m S σ \nabla u d 𝔰 . \end{array}

Thus from equations (8) and (9)

\begin{array}{l} 4 π \dot{σ} = - 2 π \nabla · σ^{2} + \nabla ∭ S \nabla u (V σ τ - m σ) d 𝔰 + \nabla ∭ u d (m d 𝔰) ∕ d t, & (10) \end{array}

and

\begin{array}{l} 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 . & (11) \end{array}

Sir Wm. Thomson’s theory

Sir Wm. Thomson’s Theory

91. We are now in a position to examine the two theories. We ﬁrst take Thomson’s, which is considerably the simpler, and which therefore serves as an introduction to the other. We have then $m = 0$ . Thus equations (10) and (11) become

\begin{array}{l} 4 π \dot{σ} = - 2 π \nabla · σ^{2} + \nabla ∭ S σ τ \nabla u d 𝔰, & (12) \\ p ∕ D - σ^{2} ∕ 2 + {(4 π)}^{- 1} ∭ S σ τ \nabla u d 𝔰 = H, & (13) \end{array}

for in the present theories we may put $v = 0$ .

We shall consider the two terms on the right of equation (12) separately. The second term gives then an apparent force per unit mass due to a potential

\begin{array}{l} - {(4 π)}^{- 1} ∭ S σ τ \nabla u d 𝔰 . & (14) \end{array}

Comparing this with equation (33) Section 14 above we see that this potential is the same as that of a magnetic system given by

\begin{array}{l} 4 π I = V σ τ . & (15) \end{array}

Now this magnetism is zero where $τ$ is zero. In other words it is only present where the vortex-atoms are, and therefore it cannot be so distributed as to give an apparent force of gravitation, for taking the view of magnetic matter expressed in equation (34) Section 14 we see that there must, in every complete vortex-atom, be a sum of magnetic matter exactly $= 0$ . Nor again is it likely to explain the phenomena of permanent magnets, because, assuming that for any given small space including many vortex-atoms $∭ V σ τ d 𝔰$ is not zero, the apparent force produced will aﬀect all other parts of space independently of whether this same integral for them is not or is zero. But to explain the phenomena of permanent magnets we must assume that the eﬀect takes place only on portions of space where there is positive magnetic matter. This term then gives us no phenomena analogous to physical phenomena. As a matter of fact it probably has no visible eﬀect on large groups of vortices, for there is no reason to suppose that the vector $V σ τ$ is distributed otherwise than at random.

92. Let us now consider the other term in equation (12), and neglecting the term already considered, put

\begin{array}{l} \dot{σ} = - \nabla · σ^{2} ∕ 2 . & (16) \end{array}

The phenomena resulting from this are the same as would follow from a stress in a medium, the stress being an equal tension in all directions $= - σ^{2} ∕ 2$ . Now comparing equation (62) Section 14 with equation (4) Section 25, we see that $σ$ depends on $τ ∕ 4 π$ in exactly the same way as does $H$ on $C$ . The question then arises—is the stress we are now considering equivalent in its mechanical eﬀects upon the vortex-atoms to the stress given by equation (68) Section 14 above, which explains the mechanical eﬀect of one current on another? We saw in Section 14 that this stress is a tension $- H^{2} ∕ 8 π$ along the vector $H$ and an equal pressure in all directions at right angles. The eﬀects then would be the same only if $σ$ be at right angles to the surface of our atom. But this is obviously not in general the case. From this analogy we can see however what approximately will happen to our atom. For instead of $σ$ being at right angles to the surface it is in all probability very nearly tangential. Assuming that it is actually tangential we see that at the surface of the atom we have a tension exactly corresponding to the pressure which in the electric analogue will be exerted on this surface. In other words, the atoms will act on each other very approximately in what may be called a converse way to the small circuits in the electric analogue; i.e. where, in the electric analogue there is an attraction, in the hydrodynamic case there will be an apparent repulsion, and vice versâ.

Now each vortex-atom forms a small circuit and therefore acts in the converse way to a small magnet. In other words, each atom acts upon each other atom as if it were charged with attracting magnetic matter. Thus we see that if we could suppose certain extra atomic vortices to exist and to be disposed throughout space in what at present must be considered quite an artiﬁcial manner with reference to the atomic vortices we could rear up a fabric which would explain gravitation. This conception however is of very little use for our present purpose.

From these considerations I think we have every reason to believe that Thomson’s theory in its native simplicity does not promise to lead us to the physical phenomena of matter. We pass on therefore to Hicks’s.

Prof. Hicks’s theory

Prof. Hicks’s Theory

93. Hicks in his theory, as I understand him, assumes that the bubbles always remain associated with the same particles of the ﬂuid. This of course is probably not the case. By reason of the variation of the pressure with the time it is probable that evanescent bubbles start into existence and disappear at various parts of the ﬂuid. This requires some few preliminary remarks.

The particles of the ﬂuid with which bubbles are permanently (i.e. throughout the greater part of each small but not inﬁnitely small interval of time) associated are those where the intensity of spin is greatest. If the intensity of spin is quite various at diﬀerent points we shall thus have vortices where there is generally no bubble, extra material vortices in fact. We must suppose these distributed quite at random till the more exact mathematical treatment of our problem leads us to suppose otherwise. Now evanescent bubbles will occur rather in these vortices than in parts of the ﬂuid where is no vortex at all (if we may suppose such parts to exist), and of course they will occur more readily in stronger than in weaker vortices. At the present stage of the theory then we may suppose evanescent bubbles to occur in all parts of the ﬂuid. As a ﬁrst approximation to the consideration of the eﬀect of these bubbles we may assume a part $m'$ of $m$ to be continuously distributed through space. Putting

\begin{array}{l} m = m' + M, & (17) \end{array}

we must suppose $M$ to be present only at the material bubbles where it is probably discontinuous, whereas $m'$ is continuous throughout both the material vortices and the rest of the ﬂuid.

Now when on account of variation of pressure $m'$ and $M$ are aﬀected—is it probable that in the neighbourhood of a permanent bubble $m'$ and $M$ are of the same or opposite signs? To answer this question observe that what we call $m'$ continuously distributed is really a series of discontinuous values of $m$ scattered at random through space so that $m'$ is probably very small. A decrement of pressure will cause an increment of evanescent bubbles, i.e. a decrement in $m'$ . An increment of the permanent bubbles will also take place the magnitude of which by what has been said concerning $m'$ will not be by any means accounted for by the decrement in $m'$ . There will therefore also be a decrement in $M$ . Similarly for an increment in the pressure. $M$ and $m'$ may therefore be assumed to be of the same sign.

After noticing that $m'$ is continuous and therefore that there is no objection to introducing its space variations we are furnished with all the materials necessary for discussing our problem. The equation we shall use is (10) of Section 25 above. We divide its discussion into two parts as follows.

26. Consideration of all the terms except $- \nabla · (σ^{2}) ∕ 2$

94. The reasons that we have already seen in Section 25 for neglecting $\nabla ∭ S σ τ \nabla u d 𝔰$ still hold good so we put this aside. This is not the case with $- \nabla ∭ m S σ \nabla u d 𝔰$ but we can neglect $\nabla ∭ u d (m d 𝔰) ∕ d t$ for the average value of $d (m d 𝔰) ∕ d t$ for any particle is zero. The only term to consider then is $- \nabla ∭ m S σ \nabla u d 𝔰$ . Putting as in equation (17) $m = m' + M$ we have

\begin{array}{l} - \nabla ∭ M S σ \nabla u d 𝔰 - \nabla ∭ m' S σ \nabla u d 𝔰 . \end{array}

The ﬁrst term of this can be neglected for the same reasons as for neglecting that containing $V σ τ$ . Applying equation (9) Section 4 to the last term and neglecting the surface integral as usual we get

\begin{array}{l} \nabla ∭ u S \nabla (m' σ) d 𝔰 = \nabla ∭ u {m' (m' + M) + S σ \nabla m'} d 𝔰 . & (18) \end{array}

The last term can probably be neglected though we cannot give such good reasons as for the other terms we have neglected. At any rate in places not near permanent bubbles $S σ \nabla m'$ is as likely to be positive as negative and vice versâ, so that such portions of space will on the whole produce no eﬀect on the permanent bubbles. If $S σ \nabla m'$ contributes anything for parts of space in the neighbourhood of permanent bubbles we must be content at present with the assumption either that the contribution is in general positive or that if it be negative it is not suﬃcient to cancel the eﬀect of the positive term $m' M$ . Remembering that $m'$ is small compared with $M$ we are left with the positive term $m' M$ . This as can be easily seen from the form of equation (18) leads to an apparent law of gravitation for our permanent bubbles.

95. The gravitational mass which we must on this supposition assign to each permanent bubble varies as the average value of $m' M$ for that bubble⁴⁴. Now we saw in Section 23 that the probable measure of mass of a permanent bubble was proportional to its average size. Do these two results agree? I cannot say, but even if they do not these considerations would still explain the motions of the solar system, but if the sun and Jupiter (say) were to collide their subsequent motion would not be that due to the collision of two bodies the ratio of whose masses is that which is accepted as the ratio of the sun’s and Jupiter’s. As a matter of fact however I should imagine that the average value of $m' M$ for a permanent bubble is proportional to its average volume and this simply as a consequence of the reasoning in Section 23 above.

A conclusion at any rate to be drawn from the above is that there is a presumption in favour of Hicks’s theory explaining gravitation.

27. Consideration of the term $- \nabla · (σ^{2}) ∕ 2$

96. In considering this term we adopt the method of Section 25 and consider an electric analogue. The analogue is an electro-magnetic ﬁeld for which in the notation of Section 14 to Section 14 above at every point,

\begin{array}{l} H = σ . & (20) \end{array}

For this ﬁeld we have at once

\begin{array}{l} 4 π C = V \nabla H = τ . & (21) \end{array}

The distribution of the magnetism in the ﬁeld is somewhat arbitrary, but in the notation of equations (61) and (62) Section 20 it will be found that everything is satisﬁed by putting

\begin{array}{l} 4 π I = - \nabla S q . & (22) \end{array}

This gives as it should $S \nabla (H + 4 π I) = 0$ , which is in fact the only equation it is necessary to satisfy. We have further

\begin{array}{l} B = H + 4 π I & = σ - \nabla S q = \nabla V q, \\ so that & A & = V q . & (23) \end{array}

Thus all the important vectors in the analogue are determined. It remains to compare the mechanical eﬀects of the analogue with the term $- \nabla · σ^{2} ∕ 2$ .

$I$ , it must be observed, is not conﬁned to the bubbles, but is distributed throughout space.

97. If we now assume that bubbles have not always existed in the positions which we call permanent, there cannot at the surface of the bubbles be any circulation round them. This makes the velocity at the surface almost normal to it, so that the stress given in equation (66) Section 14 reduces to a tension $- S H (2 B - H) ∕ 8 π$ over the surface, i.e. we have a pressure

\begin{array}{l} - σ^{2} ∕ 8 π + S σ \nabla V q ∕ 4 π . \end{array}

Now on account of the absence of circulation $\nabla V q$ is very small and may therefore be neglected. Thus we get a pressure $- σ^{2} ∕ 8 π$ , and are thus led once more to a “converse” of the analogue. This at once⁴⁵ leads to another reason for the law of gravitation if the pulsations are synchronous. This we have already seen to be probable.

The present consideration of the subject is merely to point to a method of investigating the theory of vortex-atoms. I therefore leave the subject here, not attempting to force the phenomena we have been considering to tally with the known phenomena of electricity and magnetism. Nevertheless I may say that the prospect of discussing these things by means of the present subject can scarcely be considered as distant after what has gone before.

To sum up, this ﬁrst application of the method leads to a presumption in favour of Hicks’s theory leading to an explanation of both the important properties of matter—inertia and the law of gravitation—and there is also reason from it to hope that the phenomena of electro-magnetism are not unlikely to receive an explanation. Thomson’s theory on the other hand would seem to fail in the ﬁrst two at any rate of those endeavours.

98. We close the essay with the fulﬁlment of the promise made towards the end of Section 24. In that section it will be remembered we considered a hypothetical ﬂuid for which $D = tanh (p ∕ c)$ , and made this do duty for a liquid containing bubbles. Strictly speaking our liquid is bounded at the bubbles and therefore as a bounded liquid should it be treated. For such a liquid we require an equation corresponding to equation (10) Section 25, and if possible equation (11) also. This last I have been unable to obtain, and I am not sure that to solve the problem explicitly is possible.

Our problem only deals with an incompressible ﬂuid, but as the removal of this restriction does not greatly complicate the work we will consider the general case of a bounded compressible ﬂuid. We have

\begin{array}{l} 4 π σ & = \nabla^{2} ∭ u σ d 𝔰 = - \nabla ∭ \nabla u σ d 𝔰 \\ = \nabla ∭ u (τ + m) d 𝔰 - \nabla \iint u d Σ σ, \end{array}

whence

\begin{array}{l} 4 π \partial σ ∕ \partial t = \nabla ∭ d {u (τ + m) d 𝔰} ∕ d t - \nabla \iint d (u d Σ σ) ∕ d t . \end{array}

The justiﬁcation of using $\partial ∕ \partial t$ on the left and $d ∕ d t$ under the integral sign will appear if the increment $(\partial σ ∕ \partial t) d t$ in $σ$ at a given point in the time $d t$ be considered. It will be observed that the meaning here to be attached to $\dot{u}$ will be $- S σ \nabla u$ , as in the diﬀerentiation $\partial ∕ \partial t$ with regard to the time the origin of $u$ is assumed to be ﬁxed.

I shall now merely indicate the method of procedure. By the method exhibited in Section 25 we can prove that

\begin{array}{l} d (u τ d 𝔰) ∕ d t & = - V \nabla u V σ τ d 𝔰 - u σ S τ Δ d 𝔰 \\ and that \\ d (u m d 𝔰) ∕ d t & = - m S σ \nabla u d 𝔰 + u d (m d 𝔰) ∕ d t . \end{array}

From this we can deduce that

\begin{array}{l} 4 π \partial σ ∕ \partial t & = \nabla V \nabla ∭ u V σ τ d 𝔰 + \nabla ∭ {- m S σ \nabla u d 𝔰 + u d (m d 𝔰) ∕ d t} \\ - \nabla \iint u σ S τ d Σ - \nabla \iint u d (d Σ σ) ∕ d t + \nabla \iint d Σ σ S σ \nabla u, \end{array}

and from this again we get

\begin{array}{l} \dot{σ} = \nabla w + \nabla w' + \nabla v, & (24) \end{array}

where $w$ and $w'$ are scalars and $v$ a vector given by

\begin{array}{l} 4 π w = - 2 π σ^{2} + ∭ {d 𝔰 S \nabla u (V σ τ - m σ) + u d (m d 𝔰) ∕ d t}, & (25) \end{array}

so that $w$ is in fact the $H - v - P$ of equation (11) Section 25

\begin{array}{l} 4 π w' & = \iint (S d Σ σ S σ \nabla u - u d S d Σ σ ∕ d t), & (26) \\ 4 π v & = \iint (- u σ S τ d Σ + V d Σ σ S σ \nabla u - u d V d Σ σ ∕ d t) . & (27) \end{array}

This last equation may be put in what for our purposes is the more convenient form

\begin{array}{l} 4 π v = - \iint {u σ S τ d Σ + d (u V d Σ σ) ∕ d t} . & (28) \end{array}

Again it may be put in a form free from $d ∕ d t$ ; for $V d Σ \dot{σ} = 0$ because the surface is a free surface and $d (d Σ) ∕ d t = \nabla_{1} S d Σ σ_{1}$ ⁴⁶, as can easily be proved by considerations similar to those in Section 21. Thus

\begin{array}{l} 4 π v = \iint (- u σ S τ d Σ + V d Σ σ S σ \nabla u - u V \nabla_{1} σ S d Σ σ_{1}) . & (29) \end{array}

In the problem we have to discuss $m = 0$ , so that $w$ gives only terms which we discussed in considering Sir Wm. Thomson’s theory. Observing that if in $w$ we change $m d 𝔰$ into $- S d Σ σ$ we get for the part of $w$ containing $m$ , $w'$ ; we see that $w'$ only gives terms that we have virtually discussed under Hicks’s theory. We have however entirely neglected $v$ . Are we justiﬁed in this? In the ﬁrst place we have seen that if the bubbles have not always been associated with those parts of the ﬂuid with which they now are there is round every bubble absolutely no circulation. This shows that for any one bubble $\iint V d Σ σ = 0$ ⁴⁷, and therefore we are justiﬁed in neglecting the last term in equation (28). We are probably also justiﬁed in neglecting the ﬁrst term, for probably $τ$ is very nearly tangential to the surface and therefore $S τ d Σ = 0$ .

[Note added, 1892. The whole of this last section is in rather a nebulous stage, and since writing it I have not had suﬃcient leisure to return to the matter. I hesitated whether to include it in the present issue. But since, notwithstanding the absence of any reliable results, it serves very well to illustrate how investigations are conducted by Quaternions, I have thought it worth publishing.

Should anybody feel inclined to attempt to apply the method or an analogous one it is well to note that in the Phil. Mag. June, 1892, p. 490, I have given the more general result sought in this last section. As the result is not there proved I give one proof which seems instructive. It exhibits the great variety of suitable quaternion methods of dealing with physical questions. It furnishes incidentally a fourth quaternion proof of the properties of vortices. It also illustrates how special quaternion methods developed for use in one branch of Physics at once prove themselves useful in other branches.

Adopting the notation and terminology of Phil. Trans. 1892, p. 686, §§ 5–7, let $σ$ and $τ$ be taken as an intensity and ﬂux respectively, $τ$ still being $= V \nabla σ$ and therefore $τ' = V \nabla' σ'$ . Thus $σ'$ and $τ'$ are the actual velocity and double spin respectively and the equation of motion is

\begin{array}{l} \dot{σ}' = - \nabla' (v + P) . \end{array}

Putting $σ' = {χ'}^{- 1} σ$ and operating on the equation by $χ'$

\begin{array}{l} \dot{σ} + χ' d ({χ'}^{- 1}) ∕ d t · σ = - \nabla (v + P) . \end{array}

Now

\begin{matrix} χ' d ({χ'}^{- 1}) ∕ d t · σ = - \dot{χ}' {χ'}^{- 1} σ = - \dot{χ}' σ' = \nabla_{1} S σ' σ' = \nabla · {σ'}^{2} ∕ 2 . \\ ∴ \dot{σ} + \nabla · {σ'}^{2} ∕ 2 = - \nabla (v + P), & (A) \end{matrix}

which can easily be deduced from or utilised to prove Lord Kelvin’s theorem concerning “ﬂow,” equation (23) Section 17 above.

As $d ∕ d t$ is commutative with $\nabla$ , $\dot{τ} = 0$ or $τ$ is an absolute constant for each element of matter. This being interpreted at once gives the well-known properties of vortices in their usual form.

If in the equation $4 π (v + P) = S · \nabla^{2} ∭ u (v + P) d 𝔰$ we carry one $\nabla$ across the integral sign, get rid of its diﬀerentiations which aﬀect $u$ by equation (9) Section 4 above, and then do the same with the other $\nabla$ we get

\begin{array}{l} 4 π (v + P) = \iint {(v + P) S d Σ \nabla u - u S d Σ \nabla (v + P)} + ∭ u \nabla^{2} (v + P) d 𝔰 . \end{array}

At surfaces of discontinuity in $σ$ , $v$ and $P$ will both be continuous, so that instead of $\iint (v + P) S d Σ \nabla u$ we may write $\iint_{b} (v + P) S d Σ \nabla u$ . In the last equation substitute throughout for $\nabla (v + P)$ from equation ( $A$ ). Thus

\begin{matrix} \begin{aligned} 4 π (v + P) = \iint_{b} (v + P) S d Σ \nabla u & + \iint u S d Σ \dot{σ} - ∭ u S \nabla \dot{σ} d 𝔰 \\ + (\iint u S d Σ \nabla · {σ'}^{2} ∕ 2 - ∭ u \nabla^{2} {σ'}^{2} d 𝔰 ∕ 2) \end{aligned} \\ = \iint_{b} (v + P) S d Σ \nabla u + \iint u S d Σ \dot{σ} - ∭ u S \nabla \dot{σ} d 𝔰 + ∭ S \nabla u \nabla · ({σ'}^{2}) d 𝔰 ∕ 2, \end{matrix}

which is equation (36) of the Phil. Mag. paper.

If the standard position and present position of matter coincide it is quite easy to prove that

\begin{array}{l} d S d Σ' σ' ∕ d t & = S d Σ (\dot{σ} + V \nabla_{1} σ σ_{1}) \\ d (S \nabla' σ' d 𝔰') ∕ d t & = S \nabla (\dot{σ} + V \nabla_{1} σ σ_{1}) d 𝔰 . \end{array}

Substituting for $S d Σ \dot{σ}$ , $S \nabla \dot{σ}$ from these in the last equation we get equation (32) of the Phil. Mag. paper; but this equation can also be proved directly. It should be noticed that equations (34) and (35) of that paper have been wrongly written down from equation (32). In each read $+ \nabla · (σ^{2} ∕ 2)$ for $- \nabla · (σ^{2} ∕ 2)$ .]

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