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Font size abscissae are z and z-\-dz; the ordinates will be w and

w+ -j-dz.

After the time t these same ordinates will belong to points
whose abscissae will have become (in fig. 2) z+wt and

z + dz+ (w+-^ dzfi.

Hence the horizontal distance between the points, which was
dz, will have become

i'^i'h

and therefore the tangent of the inclination, which was-^,

will have become

dw
d^

dw_^

(A.)

At those points of the original curve at which the tangent
is horizontal, â€” =0, and therefore the tangent will con-
stantly remain horizontal at the corresponding points of the
altered curve. For the points for which -r- is positive, the

denominator of the expression (A.) increases with t, and there-
fore the inclination of the curve continually decreases. But

when ^ is negative, the denominator of (A.) decreases as t

increases^ so that the curve becomes steeper and steeper. At
last, for a sufficiently large value of /, the denominator of (A.)
becomes infinite for some value of z. Now the very for-
mation of the differential equations of motion with which we
start, tacitly supposes that we have to deal with finite and con-
tinuous functions ; and therefore in the case under considera-
tion we must not, without limitation, push our results bevond
the least value oft which renders (A.) infinite. This value is
evidently the reciprocal, taken positively, of the greatest ne-

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352 Mr. G. G. Stokes on a difficulty in the Theory of Sound.

din
gative value of -r- ; "jd here, as in the whole of this paragraph,

denoting the velocity when /=sO.

By the term continuous Junction^ I here understand a func-
tion whose value does not alter per saltum^ and not (as the term
is sometimes used) a function which preserves the same alge-
braical expression. Indeed, it seems to me to be of the ut-
most importance, in considering the application of partial
diiferential equations to physical, and even to geometrical
problems^ to contemplate functions apart from all idea of alge-
braical expression.

In the example considered by Professor Challis,

w=Â»i sin ^ {xrâ€” (a + tt?)^},

A

where m may be supposed positive ; and we get by differen-
tiating and putting t^O^

dw 2iri?i 2t*
-J- = â€”-â€” cos - -,
dz X A

the greatest negative value of which is â€” ; so that the

greatest value oft for which we are at liberty to use our results

without limitation is - â€” , whereas the contradiction arrived at

by Professor Challis is obtained by extending the result to a

larger value of ^, namely â€” .

Of course, after the instant at which the expression (A.)
becomes infinite, some motion or other will go on, and we
might wish to know what the nature of that motion was. Per-
haps the most natural supposition to make for trial is, that a
surface of discontinuity is formed, in passing across which there
is an abrupt change of density and velocity. The existence
of such a surface will presently be shown to be possible, on
the two suppositions that the pressure is eaual in all directions
about the same point, and that it varies as tne density. I have
however convinced myself, by a traui of reasoning which I do
not think it worth while to give, inasmuch as the result is
merely negative, that even on the supposition of the existence
of a surface of discontinuity, it is not possible to satisfy all the
conditions of the problem by means of a single function of the
form/{aâ€” (a+w)/}. Apparently, something like reflexion
must take place. Be that as it may, it is evident that the
change which now takes place in the nature of the motion,
b^inning with the particle (or rather plane of particles) for

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Mr. G, G. Stokes on a difficulty in the Theory of Sound. 353

which (A.) first becomes infinite, cannot influence a particle at
a finite distance from the former until after the expiration of
a finite tune. Consequently, even after the change in the
nature of the motion, our original expressions are applicable,
at least for a certain time, to a certain portion of the fluid. It
was for this reason that I inserted the words '^ without limita-
tion,'' in saying that we are not at liberty to use our original
results without limitation beyond a certain value of t. The
full discussion of the motion which would take place after the
change above alluded to, if possible at all, would probably
require more pains than the result would be worth.

I proceed now to consider the possibility of the existence
of a surface of discontinuity, and the conditions which roust
be satisfied at such a surface. Although I was led to the
subject by considering the interpretation of the integral (1.),
the consideration of a discontinuous motion is not here intro-
duced in connexion with that interpretation, but simply for its
own sake ; and I wish the two subjects to be considered as
quite distinct.

Suppose that in passing across a point Q in the axis of z
the velocity and density change suddenly from u?, ^ to tx/, p',
and let b be the velocity of propagation of the surface of dis*
continuity. Let us first investigate the equation which ex-
presses that there is no generation or destruction of mass at
the surface of discontinuity, the equation in fact which takes
the place of the equation ofcontinuity, which has to be satisfied
elsewhere*

Take two points A, B in the axis of z, the first on the ne*
gative and the second on the positive side of Q, and let QA=A)
QB=A'. Take also QQ^^ndt^ so that Q' is the point where
the surface of discontinuity cuts the axis of 2;atthetime^ + ^/Â«
The quantities A, K are supposed to be very small, and will
be maae to vanish after QQ'. Consider the portion of space
comprised within a cylinder whose ends consist of two planes,
of area unity, drawn through the points A, B perpendicular
to the axis. In the time dt^ the mass of fluid which flows in
at the plane A is ultimately fnodt^ and that which flows out at
the plane B is ultimately ffnddt. Hence the gain of mass
within the cylinder is ultimately ifmâ€”ff^dt. Now the mass
at the time t is ultimately ph-\-f}vy and that at the time^ + ^f/is

{f+%dt){h+^dt)+{f^^^di){K-Â»dty,

and therefore the gain of mass is ti{p^of)dt, h and U being
omitted, since they are to be made to vanisn in the end. Equa*
ting the two expressions for the gain of mass^ we get

f>wâ€” p'ii/=(pâ€” p')o (2.)

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354, Mr. G. G. Stokes on a difficulty in the Theory of Sound.

It remains to form the equation of motion. Now we have
moving force on plane A=a^p, ultimately ;
moving force on plane B=ay, ultimately;
/. resultant moving force =a*(pâ€” p'),
â€˘'â€˘momentum generated in time d/=Â«*(pâ€” p')ÂŁf^

Now the momentum of the mass contained within the cylinder
at the time i is ultimately pwh+^'tK/h^ and the momentum of
the same set of particles at the time t + dt is

and therefore the gain of momentum is ultimately

{ (pw-p'u/) Â» -ptoÂ« +^'tt/Â«} A;
whence we have

(pw-p'tt/)Â»-(/9tt^-p'tt/Â«)=aÂ«(p-p'). . , â€˘ (3.)
By eliminating o between (2.) and (9.)> we get

(w/-w)V=Â«'(p-pO*, .... (4.)

an equation which we roav if we please employ instead of (5.).

The equations (2.), (S.) being satisfied, it appears tliat the
discontinuous motion is dynamically possible. This result,
however, is so strange, that it may be well to consider more
in detail the simplest possible case of such a motion*

Conceive then an infinitely long straight tube filled with air,
of which the portion to the left of a certain section Â« is of a
uniform density p, and at rest, while the portion to the riffht
is of a uniform smaller density p', and is moving in the positive
direction with a uniform velocity Â«/, the surface of separation
s at the same time travelling backwards into the first portion
with the uniform velocity o . The conception of such a motion
havinff been formed, consider next whether the motion is
possible or impossible ; that is to say, not whether it is pos-
sible or impossible in the actual state of elastic fluids, but
whether it would or would not be consistent with dynamical
principles in the case of an ideal elastic fluid, in which the
pressure was equal in all directions about the same point, and
varied as the density.

In the case under consideration, the fluid to the left of s is
in equilibrium in the simplest way. The fluid to the right is
of uniform pressure, and there is no generation or destruction
of velocity. The only question^ then, can be as to the possi-
bility of the passage from the one state into the other taking
place in the way supposed. In the first place, it is evident
that, independently of any consideration of force, there must
be a relation between p, p', tc/, and o ; for a length o / of con-

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Mn G. G. Stokes on a difficulty in the Theory of Sound. S65

densed air comes to occupy, in the rarefied state, a length
(a+tt/y, so that we must have po=p'(b-f Â«/). Next, if we
take two sections s^j s<^ the first to the left and the second to
the right of 5, and suppose the first to remain at rest, with the
fluid in which it is situated, while the second moves, along
with the fluid in which it is situated, with the velocitv u/, since
the pressure on s^ exceeds that on 5, by a^(pâ€” p'), the surface
of 5, or 8^ being for simplicity's sake supposed equal to unity,
there must in the time / be generated a momentum a^(pâ€” p')^
in the fluid lying between 5^ and 5o. But this will be the case
in consequence of the velocity w being communicated to a
volume 0/ of air which was previouslv at rest and of density p,
while the state of rest or motion of the remainder of the air
between s^ and s^ has been unaltered, provided a^{p^p^^*aftip.
These two relations being satisfied, it appears that the motion
is dynamically possible. The two equations might have been
obtained at once fi-om (2.) and (3.) by writing â€” b for o and
putting TO =0, but I have preferred deducing them afresh firom
first principles in consequence of the novelty of the subject,
and the reluctance with which the conclusions that I have
arrived at are likely to be received by mathematicians.

These conclusions certainly seem sufficiently startling ; yet
a result still more extraordinary remains behind. By solving
tlie two equations of the preceding paragraph with respect to
Â«/ and 0, we get

^â€˘^ "=\^Â« -

Nowletp' vanish; then'o/becomes infinite and evanishes. Hence
the rate at which the condensed air (which remains packed
like the combustible matter in a rocket) is discharged decreases
indefinitely as the space into which the discharge takes place
approaches indefinitely to a vacuum. Of course the velocity
of discharge becomes infinite, without which the requisite mo-
mentum could not be furnished. The quantity of air which
passes in a unit of time across a plane, of area unity, taken at
the positive side of the tube, is wp\ which is easily seen to be
a maximum, for a given value of p, when p^=jf'

A similar paradox is fully considered by MM. Barrfi de

* It is worthy of remark, that when p' is very nearly egual to p, and
consequently u/ very small, the velocity of propagation is very nearly
e^ual to a, to which it approaches inddKnitely when w' is indefinitely dimi-
nished. Thus even this discontinuous motion offers no exception to the
theorem, at once proved by neglecting the squares of small quantities, that
in very small motions any disturbance is propagated in the fluid with the
velocity a.

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356 Mr. G. G. Stokes on a diffictdty in the Theory of Sound.

Saint- Venant and Wantzel, in the 27th Cahier of the Journal
de FEcole Polytechnique.

The strange results at which I have arrived appear to be
fairly deducible from the two hypotheses already mentioned.
It does not follow that the discontinuous motion considered
can ever take place in nature, for we have all along been rea-
soning on an ideal elastic fluid which does not exist in nature.
In the first place, it is not true that the pressure varies as the
density, in consequence of the heat and cold produced by con-
densation and rarefaction respectively. But it will be easily
seen that the discontinuous motion remains possible when we
take account of the variation of temperature due to condensa-
tion and rarefaction, neglecting, however, the communication
of heat from one part of the fluid to another. Indeed, so far
as the possibility of discontinuity is concerned, it is immaterial
according to what law the pressure may increase with the
density.

Of course the communication of heat from one particle of
the fluid to another would affect the result, though whether
to the extent of preventing the possibility of discontinuity I
am unable to say. But there is another supposition that we
have made which is at variance with the actual state of elastic
fluids. It is not true that one portion of an elastic fluid is
incapable of exerting any tangential force on another portion
on which it slides, even though the variation of velocity from
the one portion to the other be not abrupt but continuous.
In consequence of this tangential force,* analogous in some
respects to friction in the case of solids, the mutual pressure
of two adjacent elements of a fluid is not accurately normal to
the surface of separation, nor equal in all directions about the
same point. In many cases the influence of this internal fric-
tion is insensible, while in other cases it is very important
Its general effect is to check the relative motion of the parts
of a fluid. Suppose now that a surface of discontinuity is very
nearly formed, that is to say, that in the neighbourhood of a
certain surface there is a very rapid change of density and
velocity. It may be easily shown, that in such a case the
rapid condensation or rarefaction implies a rapid sliding mo-
tion of the fluid ; and this rapid sliding motion would call into
play a considerable tangential force, the effect of which would
be to check the relative motion of the parts of the fluid. It
appears, then, almost certain that the internal friction would
effectually prevent the formation of a surface of discontinuity,
and even render the motion continuous again if it were for an
instant discontinuous.

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[ 357 ]

LV. On the Spontaneous Cohesion of the Particles of
Alumina. By Richard Phillips, F.R.S.^

IT has been observed by M. Wittstein (Chemical Gazette,
April 1847), that the precipitate which is obtained from
the persulphate or perchloride of iron, if kept for a great length
of time in water, loses almost entirely the property of being
soluble in acetic acid. This statement recalled an observation
which I made several years since with respect to the similar
cohering power of alumina; the two cases differing, however,
in these respects, that whereas the sesquioxide of iron requires
one or probably two years for the production of the effect,
alumina undergoes the change partially in a very short time ;^
the precipitated alumina does not, however, assume a crystal-
line appearance, as stated by M. Wittstein to be the case with
the cohering sesquioxide of iron.

It is well known, both with respect to recently precipitated
sesquioxide of iron and alumina, tliat they are soluble even in
acetic acid, and consequently in the more powerful acids ; but
if the precipitated alumina be kept for two days even moist,
and in contact with the solution from which it was precipitated,
I find that sulphuric acid does not immediately dissolve it.

The following experiment will perhaps elucidate the fact
which I have now stated : I precipitated the alumina from a
large quantity of alum by excess of ammonia, and after keep-
ing it moist for fourteen days, I washed it to get rid of the
excess of ammonia and the salts existing in solution ; to a
quantity of this precipitated alumina I added dilute sulphuric
acid in excess, and almost immediately separated by the filter
the insoluble from the dissolved portion ; the latter, being
washed and ignited, weighed twenty and a half grains. The
sulphuric solution of the soluble portion was decomposed by
ammonia, and weighed, after washing and igniting, eight
grains ; so that in about a fortnight, nearly 72 per cent, of the
particles of the alumina had so cohered as to resist the im-
mediate chemical action of the sulphuric acid. I did not try
whether solution would have been efiected by longcold digestion
in this case, but I have observed that time greatly increases the
solvent power of the acid ; I found, however, in the present
instance, that the alumina which cold dilute acid would not im-
mediately dissolve was entirely soluble in it when boiling ; but
if my recollection serves, I have found that, by long keeping,
the alumina has become insoluble even in boiling sulphuric
acid.

When, however, a solution of alum is decomposed by the

â€˘ Read before the British Association held at Swansea, August 9, 1848,
and communicated by the Author.

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358 Mr. R. Phillips on the Spontaneous Cohesion of

carbonate of soda or the sesquicarbonate of ammonia, the
result as to the solubility of the precipitated alumina is totally
different; the precipitate in the latter case is more dense^and
instead of being semitransparent and gelatinous, it is opake
and pulverulent, these differences evidently denoting some
variation of constitution. Accordingly, I find that whatever
may liave been the length of time which I have kept this pre-
cipitate moist, I have always found it totally and immediately
soluble in cold dilute sulphuric acid*

If it were not almost universally admitted that carbonic acid
does not combine with alumina, the most obvious conclusion
to arrive at would be that carbonate of alumina is actually
formed, and that the carbonic acid preventing the cohesion
of the particles of alumina, it remained soluble in acid*

I shall now, however, state some experiments which seem
to me to prove that carbonate of alumina may be formed. I
precipitated 200 grains of alum by excess of sesquicarbonate
of ammonia, and washed the precipitate long after it ceased to
render the water alkaline. After the alumina had been pre-
cipitated about fourteen days, I dissolved it in a counterpoised
vessel in dilute sulphuric acid ; solution took place with effer-
vescence, and twenty grains of carbonic acid gas were evolved.
It follows therefore, I think, that carbonate of alumina was
formed, and that the carbonic acid interposed to prevent the
cohesion of the alumina.

A very moderate heat seems, however, sufficient to decom-
pose carbonate of alumina. A quantity of precipitate, pre-
pared as above stated by sesquicarbonate of ammonia, was
dried by the heat of steam ; a portion of this was added to
dilute sulphuric acid, and although effervescence occurred and
some alumina was dissolved, the greater portion had lost its
carbonic acid and become insoluble.

I have further found, that when hot solutions of alum and
carbonate of soda are mixed, the precipitate formed is not
totally and immediately soluble in dilute sulphuric acid.

That other substances have the power of intervening to pre-
vent the cohesion of precipitated alumina is shown by the fol-
lowing experiment : â€” To a solution of alum I added one of
sulphate of magnesia, and by ammonia precipitated a mixture
of the two earths ; and on keeping the mixed precipitate for
such a length of time as would have rendered only one-fifth of
alumina precipitated per se soluble, I found that the mixed
earths were totally and almost immediately dissolved by cold
dilute sulphuric acid.

The experiments which I have now detailed prove, I think,
the following facts : â€”

1. That tne particles of alumina^ like those of peroxide of

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the Particles of Alumina. 359

iron, even when kept moist, have the power of cohering so as
to prevent chemical action.

2. That carbonate of alumina may be formed, though de-
composable at a comparatively low temperature.

3. That owing to the presence of carbonic acid or magnesia,
the particles of alumina are prevented from cohering.

It will also, I think, be admitted, {bat the cohering property
which I have now described may have led to errors in analysis,
in which precipitated alumina may have been kept without any
such intervening substances as to render it soluble. I may
remark, in conclusion, that the alumina insoluble in dilute
acids was not taken up by soda, at any rate when used cold.

Although not connected with the present subject, I will
take the opportunity of mentioning an unexpected case which
lately occurred to me. In examining a soil, I treated one
portion of it simply with muriatic acid and another with nitro-
muriatic acid. On the addition of ammonia to both solutions,
I found the muriatic solution yield a bulky precipitate, evi^
dently containing, if not consisting of protoxide of iron,
whereas the nitro-muriatic solution gave a comparatively small
quantity of precipitate evidently of sesquioxide of iron.

On investigation, I found that the protoxide of iron had
carried down with it a considerable quantity of magnesia;
whether in a state of mixture or combination remains to be
proved.

To try the experiment on a considerable scale, I mixed an
equivalent of sulphate of magnesia with half an equivalent of
protosulphate of iron, adding an equivalent of sal-ammoniac
to prevent the partial precipitation of the magnesia by the
ammonia, which I added in excess to precipitate all that it was
capable of throwing down. Having carefully washed the preci-
pitate, I redissolved it in nitro-muriatic acid, so as to peroxidize
the iron, and I again added sal-ammoniac and ammonia. By
this peroxide of iron was thrown down ; and on adding phos-
phate of soda to the filtered solution, I obtained so abundant a
precipitate of the ammonio-magnesian phosphate, as to render
it probable that at least one-fourth of the magnesia had been
precipitated with the protoxide of iron.

I may also notice another circumstance connected with the
precipitation of the magnesia with the protoxide of iron. It
is well known that excess of ammonia dissolves a portion of
protoxide of iron, which by oxidizement of the air is precipiÂ«
tated. This solution however does not happen in the case
above alluded to, the whole of the protoxide of iron being at
once precipitated with the magnesia.

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L 3G0 ]

LVI. On the Vibrations of an Elastic Fluid. By the Rev. J.
ChalliSi M.A.^ FM.S^ F.R.A.S,j Plumian Professor of
Astronomy and Experimental Philosophy in the University of
Cambridge*.

IN accordance with the intention expressed at the close of
my communication to the Philosophical Magazine for
August last, I proceed to draw some inferences from the equa-
tions (B.) and (C). But, first, it will be proper to point out
two inaccui'acies contained in that communication. The third

bH
term of equation (C.) in p. 100 should be -^ instead of Sy;

and in the same paffe, in the series for the form of f , the
quantities in brackets should be x â€” rf/ + <?!Â» Â« â€” a'/ + c^, Â« â€” a'/ + c^
&c., Cp c^ Cgi &cÂ« being certain constants. No inferences are
in the least degree affected by these alterations.

Let us now, for the purpose of testing equation (C), suppose
thatyis a function of x^+y'^. By this supposition the equa-
tion becomes, on putting r^ for ar +^^

showing that the supposition is allowable. In any case of
vibrations not very large, the second term in the brackets is
too small to have any appreciable effect. Omitting this term,

and putting ^e for -^ the integral of the resulting equation in

a series is

/=l-Â«^+ _^-p^^^+&c.

I have already stated (Phil. Mag. for April, p. 281) that
according to this result y* and ^ cannot vanish together.
Hence, since the velocity of the vibrating particle in any di-
rection perpendicular to the axis of Â« is f'-j-j and parallel to

the axis of s;,/*. -^, it follows that the fluid cannot be con-
az

stantly quiescent at a certain distance from that axis. This
conclusion is inconsistent with the non-divergence of the vibra-
tions, and would be enough to condemn the whole of the pre-
vious investigation, if the reasoning by which it was arrived at
be good.

It is now proper for me to state the grounds on which that
conclusion is to be rejected. I have maintained, and still
* Communicated by the Author.

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Prof. Challis on the Vibrations of an Elastic Fluid. 361

maintain, that to obtain true and consistent results in hydro-
dynamics another general equation, distinct from ihose com-

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