conduction and radiation. "In order then that permanency of type may be possible in a wave of longitudinal disturbance, there must be both change of temperature and conduction of heat during the disturbance.”

Rankine, in the paper referred to, considers it unlikely that the conduction of heat ever takes place in such a way as accurately to maintain the relation (43), except in the case of a wave of sudden disturbance, where we have adjacent portions of the medium at a finite interval of temperature.

This latter kind of wave is of interest because, as we have seen, any disturbance however started tends ultimately to become discontinuous. The simplest case is that in which there is no variation in the properties of the medium except at the plane of discontinuity. If p, s, u denote the values of the pressure, the bulkiness, and the particle-velocity behind, P, S, U those in front of the discontinuity, the conditions to be satisfied are obtained by making p1, s,1, u1 = p, s, u, and p2, 82, u2 = P, S, U, respectively, in the above formule. We find

and if further U=0 so that the medium is at rest in front of the

wave,

and

u = m (S-s) = ± √(p-P) (S-s),

the upper or lower sign being taken according as S is greater or less than s, i.e. according as the wave is one of sudden compression or of sudden rarefaction.

The mathematical possibility of, and the conditions for, a wave of discontinuity were first pointed out by Stokes.

CHAPTER IX.

VISCOSITY.

174. WE now proceed to take account of the resistance to distortion, known as 'viscosity' or 'internal friction,' which (Art. 2) is exhibited by all actual fluids in motion, but which we have hitherto neglected. The methods we shall employ are of necessity the same as are applicable to the resistance to distortion, known as 'elasticity,' which is experienced in the case of solid bodies. The two classes of phenomena are of course physically distinct, the latter depending on the actual changes of shape produced, the former on the rate of change of shape, but the mathematical methods appropriate to them are to a great extent identical.

175. Let Pax Pau Paz denote the components parallel to x, y, z, respectively, of the force per unit area exerted at the point P(x, y, z) across a plane perpendicular to x, between the two portions of fluid on opposite sides of it; and let Pcs Puy, P and P› P› P have similar meanings with respect to planes perpendicular to y and z respectively*. It is shewn in the theory of Elastic Solids that these nine quantities, which completely specify the state of stress at the point P, are not all independent, but that

We need not here reproduce the proof of these relations, as their truth will appear from the expressions for P., &c. to be given below.

* As is usual in the theory of Elasticity we reckon a tension positive, a pressure negative. Thus in the case of a fluid in equilibrium we have

PPP = −p.

To form the equations of motion let us take, as in Art. 6, an element dædydz having its centre at P, and resolve the forces acting on it parallel to x. Taking first the pair of faces perpendicular to x, the difference of the tensions on these parallel to x will be dpxx dx.dydz. From the other two pairs we obtain

dx

dpvz dy.dzdæ and dpt dz.dady, respectively. Hence, with our usual

dy

notation,

dz

Pat=pY+

dx

dz

.....

.....(2).

176. It appears from Arts. 2, 3 that the deviation of the state of stress denoted by Pra, Pxy, &c. from one of uniform pressure in all directions depends entirely on the motion of distortion of the fluid in the neighbourhood of P, i.e. on the quantities a, b, c, f, g, h by which this distortion was in Art. 38 shewn to be specified. Before endeavouring to express paz, Pxy, &c. as functions of these quantities, it is convenient to establish certain formulæ of transformation.

Let us draw Px', Py', Pz' in the directions of the principal axes of distortion at P, and let a', b', c' be the rates of extension along these lines. Further let the mutual configuration of the two sets of axes, x, y, z and x', y', z', be specified in the usual manner by the annexed scheme of direction-cosines.

a+b+c=a' + b' + '..........(4),

an invariant, as it should be. See Art. 8. Again

177. From the symmetry of the circumstances it is plain that the stresses exerted at Pacross the planes y'z', z'x', x'y' are wholly perpendicular to these planes. Let us denote them by P1, P2, P3, respectively. In the figure of Art. 3 let ABC be a plane drawn perpendicular to x, infinitely close to P, meeting the axes of x', y', z' in A, B, C, respectively; and let A denote the area ABC. The areas of the remaining faces of the tetrahedron PABC will then be 14, l,, Δ. Resolving parallel to x the forces acting on the tetrahedron we find

P = pl. 1 + pl. l + pl. زوا

•

2

33

the external impressed forces and the resistances to acceleration

being omitted for the same reason as in Art. 3. Hence

Pxx+Pvv+Pxx = p1 + P2 + p3 = -3p, say ...........(7),

so that p denotes the average pressure about the point P.

The truth of the relations (1) follows from (8) by symmetry. The student should notice the analogy of (3) and (5) with (6) and (8) respectively.

If in the same figure (Art. 3) we suppose PA, PB, PC to be drawn parallel to x, y, z, respectively, and ABC to be any plane near P whose direction-cosines are l, m, n, we find, in exactly the same manner, for the components of the stress exerted across this plane, the values

1Pxx + mpx + прpxx) lpvx + mpw + npvz Ipzx + mp + пр... (9), respectively.

178. Now p1, P2, p, differ from -p by quantities depending on the motion of distortion, which are therefore functions of a', b', c' only; and if a', b', c' be small we may suppose these functions to be linear.

We write therefore

p1 = − p + λ (α' + b' + c') + 2μα',

P2 = - p + λ (α' + b'+ c') + 2μδ',

P3 = − p + λ (α' + b' + c') + 2μc',

where λ, μ are constants, this being plainly the most general assumption consistent with the above suppositions, and with symmetry. Substituting these values of P1, P2, p. in (6) and (8), and making use of (3) and (5), we find

Pxx = - p + λ (a + b + c) + 2μα, &c., &c.,

Puz = 2μf, &c., &c.

But from the definition (7) of p we must have

3λ + 2μ = 0,

whence, finally, introducing the values of a, b, c, &c. from Art. 38, we have