§ 45. It is evident that a By represent a wave, or system of waves, regularly transmitted through the ether composing the common refracted light but en represent a disturbance of quite a different character, propagated with a very slow velocity, and therefore such as makes each cluster (at least, those at or near the bounding surfaces of the transparent body,) an origin of waves spreading into vacuum as if from a point, so that the bounding surfaces will appear to produce light, in the same manner as luminous surfaces.

That the natural colours of bodies, and the absorption of light by coloured media, are the effects of these waves, I hope to shew in a future paper in the following manner, viz. I shall prove that the intensity of the waves represented by en depends on the length of the waves represented by a By; and then, that the intensity of the latter waves depends in general upon the intensity of the former, and thus I shall establish a relation between the intensity of light transmitted through a medium and the length of the wave, such a relation as, I believe, is capable of accounting for the apparently irregular manner in which absorption takes place.

XXIII. On the Steady Motion of Incompressible Fluids. By G. G. STOKES, B. A. Fellow of Pembroke College.

[Read April 25, 1842.]

In this paper I shall consider chiefly the steady motion of fluids in two dimensions. As however in the more general case of motion in three dimensions, as well as in this, the calculation is simplified when udx + vdy+wdz is an exact differential, I shall first consider a class of cases where this is true. I need not explain the notation, except where it may be new, or liable to be mistaken.

To prove that udx + vdy+wdz is an exact differential, in the case of steady motion, when the lines of motion are open curves, and when the fluid in motion has come from an expanse of fluid of indefinite extent, and where, at an indefinite distance, the velocity is indefinitely small, and the pressure indefinitely near to what it would be if there were no motion.

By integrating along a line of motion, it is well known that we get the equation

=

where dV Xdx + Ydy + Zdz, which I suppose an exact differential. Now from the way in which this equation is obtained, it appears that C need only be constant for the same line of motion, and therefore in general will be a function of the parameter of a line of motion. I shall first shew that in the case considered C is absolutely constant, and then that whenever it is, udx + vdy + wds is an exact differential.

To determine the value of C for any particular line of motion, it is sufficient to know the values of p, and of the whole velocity, at

any point along that line. Now if there were no motion we should have

p. being the pressure in that case. But considering a point in this line at an indefinite distance in the expanse, the value of p at that point will be indefinitely nearly equal to p1, and the velocity will be indefinitely small. Consequently C is more nearly equal to C, than any assignable quantity: therefore C is equal to C; and this whatever be the line of motion considered; therefore C is constant.

In ordinary cases of steady motion, when the fluid flows in open curves, it does come from such an expanse of fluid. It is conceivable that there should be only a canal of fluid in this expanse in motion, the rest being at rest, in which case the velocity at an indefinite distance might not be indefinitely small. But experiment shews that this is not the case, but that the fluid flows in from all sides. Consequently at an indefinite distance the velocity is indefinitely small, and it seems evident that in that case the pressure must be indefinitely near to what it would be if there were no motion.

Differentiating therefore (1) with respect to x, we get

whence

dw

dx

dy

du

u

u

dy

dy

dv

dz

(du - dw)

+ u

v

du

dy

(dw

du

dv

w

du

du

dx

dz

du

dz

= 0.

dy

= 0,

dv du dw dv

=

=

dx dy' dy dz dz dx'

and therefore udx + vdy + wdz is an exact differential.

When udx+vdy+wdz is an exact differential, equation (1) may be deduced in another way*, from which it appears that C is constant. Consequently, in any case, udx + vdy + wdz is, or is not, an exact differential, according as C is, or is not, constant.

Steady Motion in Two Dimensions.

I shall first consider the more simple case, where uda + vdy is an exact differential. In this case u and v are given by the equations

Now from equation (3) it follows that udy - vdx is always the exact differential of a function of x and y. Putting then

dU = udy - vdx,

UC will be the equation to the system of lines of motion, C being

the parameter. U may have any value which allows

and

to

dy

dx

satisfy the equations which u and v satisfy. The first equation has been already introduced; the second leads to the equation which U is to satisfy; viz.