of Professor Rankine as to the true capacity of bodies for heat. Professor Rankine considers that the true capacity for heat of the same body can have different values when the states of aggregation of the body are different; whereas I, on the other hand, have given my reasons for supposing that the true capacity of a body for heat must be the same in all states of aggregation*.

M. de Saint-Robert now makes this same supposition, that the capacity of a body for heat is the same under all states, and consequently that the quantity of heat which a body contains is proportional to its absolute temperature; but instead of referring to the reasons which had led me to this conclusion, he merely says (page 83), "The temperature t being the outward manifestation of the quantity of heat H contained in a body under its original form of heat, it follows that whenever a body has the same temperature, it must have the same quantity of internal heat."

I cannot think that this argument will be regarded as conclusive. It does not appear to me directly evident that the outward manifestation of the heat must be the same in the different states of aggregation. If the conclusion in question could be deduced in so simple a manner, so quick-sighted a philosopher as Dr. Rankine would assuredly not maintain the opposite opinion.

V. Supplementary Researches in Hydrodynamics.-Part III. By Professor CHALLIS, M.A., F.R.S., F.R.A.S.†

THE

HE Hydrodynamical Researches communicated in the Numbers of the Philosophical Magazine for September and October 1865, which were mainly devoted to the consideration of the problem of the motions of a small sphere acted upon by the undulations of an elastic fluid, carried the solution of it so far as to evolve expressions for the acceleration of the sphere containing two undetermined arbitrary constants m1 and m'. It will be my endeavour, in continuing the Researches, to complete the solution by ascertaining the composition and values of these quantities. Having found, in the course of revising for this purpose the reasoning in the previous researches, that it requires some modifications, I shall commence with pointing these out. The novelty and the difficulty of the mathematical investigations involved in the treatment of this problem may be alleged as sufficiently explaining why I am obliged to proceed by slow and tentative steps. My reasons for considering the solution of [Professor Rankine's remarks on these observations will be found at p. 407 of the preceding Number.-ED.]

+ Communicated by the Author.

Phil. Mag. S. 4. Vol. 31. No. 206. Jun. 1866.

D

it to be absolutely necessary for the advancement of physical theory have been fully stated in previous communications.

For the better understanding of the present argument it will be proper, in the first instance, to advert briefly to several previously established theorems. One of these theorems is, that the third general equation conducts to rectilinear motion by merely supposing that uda+vdy+wdz is an exact differential. As this is an abstract analytical supposition, made without reference to any case of arbitrary disturbance, we may infer from the result that rectilinearity is a general characteristic of the motion of a fluid, in so far as the motion is not determined by arbitrary conditions. To ascertain whether the straight line along which the motion takes place is an axis relative to the contiguous motion, I have supposed that (d.fp)=udx+vdy+wdz, and that is a function of z and t, and ƒ a function of x and y, such that f=1, dx df =0, and =0, where x =0, and y=0. This supposition is justified by finding that, on introducing it into the two other general equations, an equation is obtained from them which, for points contiguous to the axis, is resolvable into the two following:

df dy

+2dp d2 dp2 d2p dt2 dz dzdt dz2 dz2

+

=0.

Of each of these equations I have obtained a particular solution— that is, a solution of explicit form to which the analysis conducts without reference to arbitrary conditions of the motion. (See an Article in the Philosophical Magazine for December 1852.) By the process referred to, the first equation gives

r being the distance from the axis of z, which is the axis of

b2

motion, and e being put for 42; and the other equation gives

for dø

dz'

or the velocity (w') along the axis,

2m2K
3a (x2-1)

cos 2qμ + &c.,

9 being put for μ for 2-at+c, and being the ratio of

w' = m sin qu

2π
λ

аф f2 do 2
+
= 0,

the velocity of propagation, to a. Also by means of the equation

is obtained, for the value of the condensation (o) along the axis,

3a2 (x2 - 1) cos 2qμ +

-

Now, since uda+vdy+wdz is strictly an exact differential only for indefinitely small distances from the axis, the above value of f is strictly true only for indefinitely small values of r; so that for the purposes of exact calculation f=1-er. On the contrary, the exact values of w' and ' are expressed by series consisting of an unlimited number of terms, each of which is a circular function of z-a1 t+c. These expressions accordingly prove that wand are propagated along the axis with the same constant velocity a1, and without undergoing alteration. The proof of this law is independent of the value of m; and whether w and of be large or small, the above series retain the same form, the terms following the first or principal terms always coexisting with the latter. For our present purpose it suffices to include only terms of the second order with respect to m, in which case x is a numerical constant, the analytical expression for which is (1+

(1+ ex2) +

Again, it is to be remarked that since explicit formulæ expressing the velocity and condensation in vibratory motion have been obtained without reference to a given mode of disturbance, in every case of disturbance producing vibrations of the fluid the motion must in a certain manner be compounded of the motions defined by these formulæ. It is therefore requisite to inquire, next, respecting the laws and the effects of this composition. From the usual approximate equations a2.do du

+ =0,
dt

a2.do dv
dt
dy

+ =0,

a2.do dw
+
dz dt

=0,

dx the known theorem that uda+vdy+wdz is approximately an exact differential for small vibratory motions may be deduced prior to the consideration of any arbitrary disturbance. Let us that in this case also that differential is equal to (d.fp), being functions of the same variables as before. Then from the above equations, and from the approximate equation of constancy of mass, viz.

suppose f and

The assumed compositions of ƒ and are satisfied by resolving

d2 d2

b2p-a2. + =0,
dz2 dt2

which are both linear with constant coefficients.

that, both in this and in the former resolution,

of

d2f

d2f
dx2 dy,

+ where x=0 and y=0. The particular solution.

of the first of these equations is the series for f already given, in which the value of r may now be taken without limitation. From the particular solution of the other equation, combined with the equations

w=f

аф
dz'

do
a2o+f =0,

we obtain

dt

The definite expressions thus arrived at apply to the principal parts of the condensation, and of the velocities parallel and transverse to the axis-that is, to the parts which are of the same order as the velocity and condensation indicated by the first terms of the series for w' and o'. It has already been argued that the parts of the velocity and condensation along the axis indicated by the other terms necessarily coexist with those that are principal; but with respect to points distant from the axis, expressions for the velocity and condensation to terms of a higher order than the first are not deducible by the preceding investigation, because the assumptions made respecting the properties of the functions f and are not satisfied beyond terms of the first order. It may, however, be assumed that the motion and condensation are symmetrically disposed about the axis, this having been shown to be exactly the case in its immediate neighbourhood, and very nearly so at all other positions. In fact, by supposing that (d¥) = (d.fp+dx)=udx+vdy+wdz, x being a function of z, r, and t, and by substituting in the general differential equation to terms of the second order, of which is

I have found that the equation is satisfied if

x=m2f, sin 2q (z −кat+c),

The value of ƒ expressed in a series being known, a series for f may be readily deduced from this equation by the method of indeterminate coefficients. In this way I have obtained

There is another consideration which it will be proper to introduce in this stage of the argument. Since the differential equations which determine f and to quantities of the first order are linear with constant coefficients, it might be supposed that we may hence infer the coexistence of small vibrations. But because the value of the constant b2 is (x2-1), it fol

lows that the equation

2π

d2 b2 4-a2 dz2 d2p

λε

is satisfied by only a single value of λ, and that for every different value the equation is different. In consequence of this analytical circumstance, no general inference respecting the coexistence of small vibrations can be drawn from the above equation. In fact the foregoing investigations are only proper for determining forms of vibratory motion that are independent of arbitrary conditions. It is true that the equation might be satisfied by supposing to be the sum of any number of terms such as m sin (z-Kat+c), λ being the same for all; but in that case, as is known, the form of the sum would be the same as that of each term. It may, however, be said that in virtue of this composition the factor m might be assumed to be a particular constant, in which case the difference as to magnitude between one set of vibrations and another having the same value of λ would depend exclusively on the number of the components and their respective phases. Having premised so much as this, I proceed now to show that the velocities and condensations relative to dif

λ