formed, because in an element so easily moved as is the air small disturbances can never be entirely avoided.

It is easy to see that such an endless cylindrical jet, issuing from a tube of corresponding section into a quiet exterior fluid and everywhere containing fluid that is moving with uniform velocity parallel to its axis, corresponds to the requirements of the "steady condition."

I will here further sketch only the mathematical treatment of a case of the opposite kind, where the current from a wide space flows into a narrow canal, in order thereby also at the same time to give an example of a method by which some problems in the theory of potential functions can be solved that hitherto have been attended by difficulties.

I confine myself to the case where the motion is steady and dependent only upon two rectangular coördinates, x and y; where moreover no rotating particles are present in the frictionless fluid at the beginning, and where none such can be subsequently formed. If we indicate by u the component parallel to x of the velocity of the fluid particle at the point (xy) and by the velocity parallel to y, then, as is well known, two functions of x and y can be found such that

By these equations the conditions are also directly fulfilled that in the interior of the fluid the mass shall remain constant in each element of space, viz:

For a constant density, h, and when the potential of the external forces is indicated by V, the pressure in the interior is given by the equation

are the stream lines of the fluid, and the curves

P= constant

(16)

are orthogonal to them. The latter are the equi-potential curves when electricity, or the equal temperature curves when heat, flows in steady currents in conductors of uniform conductivity.

From the equation (1) it follows as an integral equation that the quantity + i is a function of x + yi, where i=√1. The solutions hitherto found generally express and as the sums of terms that are

themselves functions of x and y. But inversely we can consider and develop x + yi as a function of +i. In problems relative to currents between two stationary walls, is constant along the boundaries, and therefore if and are presented as rectangular coördinates in a plane, then in a strip of this plane bounded by two parallel straight lines, =co and = C1, the function x + yi is to be so taken that on the edge it corresponds to the equation of the wall, but in the interior it assumes a given variability.

A case of this kind occurs when we put

For the value± 7 we have y constant and x = Aq— Aeo.

varies from

When to the value of x changes at the same time from to - A, and then again back to -∞. The stream lines x = correspond thus to a current along two straight walls, for which y = ± Aπ and x varies between ∞ and ― A. Therefore when we consider as the expression of the stream curve the equation (2) corresponds to the flow out into endless space from a canal bounded by two parallel planes. On the border of the canal however where x-A and y = ± Aπ and where further, p=0 and 47, we have

therefore

=

Electricity and heat flow in this manner, but liquids must tear asunder. If from the border of the canal there extend stationary dividing discontinuous lines that are of course prolongations of the stream lines that follow along the wall and if outside of these discontinuous lines that limit the flowing fluid there is perfect quiet, then must the pressure be the same on both sides of these dividing lines. That is to say, along that portion of the line which corresponds to the free dividing line, in accordance with the equation (1b), we must have

=

In order now, in the solution of this modified problem. to retain the fundamental idea of the motion expressed in equation (2), we will add

to the above expression of x + y i still another term σ+7i, which is also always a function of p+ i, we have then

and must determine 6+ Ti so that along the free portion of the discontinuous surface where

= we shall have

Since is constant along the wall we can integrate the last equation with respect to p, and change the integral into a function of + i by substituting everywhere instead of p the expression +i (+7). Thus by an appropriate determination of the constants of integration we obtain

¥=± (2 a + 1) π [a being any whole number],

and

p= log 2.

Thus neither one lies between the limits from =+π to =—π.
The function σ+ri is here continuous.

If > log 2, then all these values become purely imaginary, there

0, while

ατ
dq

fore σ = has the value given above in equation (3c). This portion of the lines 7 therefore corresponds to the free portion of the jet.

If p< log 2 the whole expression is real up to the additive quantity Aiπ, which latter is to be added to the value of 7 i and y i respectively.

The equations (3a) and (3d) correspond therefore to the outflow from an unlimited basin into a canal bounded by two planes, whose breadth is 4 Aπ and whose walls extend from x=-∞ to x =—A (2—log 2). The free discontinuous line of the flowing fluid curves from the nearest edge of the opening at first a little towards the side of the positive, where for p=0, x=—A and reaches its greatest a value when y=±A 3

(7+1); then it turns inward towards the inside of the canal and at

last asymptotically approaches the two lines y=A7, so that finally the breadth of the outflowing jet is equal only to the half breadth of the canal.

1

The velocity along the discontinuous surface and at the extreme end of the outflowing jet is, so that this form of motion is possible for every velocity of efflux.

I present this example especially as it shows that the form of the liquid stream in a tube can for a very long distance be determined by the form of the initial portion.

ADDITION, BEARING ON ELECTRICAL DISTRIBUTION.

When in equation (2) we consider the quantity as the electric potential it gives the distribution of electricity in the neighborhood of the edges of two plane disks quite near together, assuming that their distance is indefinitely small with respect to the radius of curvature of their curved edge. This is a very simple solution of the problem that has been considered by Clausius.* It gives moreover the same distribution of electricity as he found for it; at least so far as it is independent of the curvature of the edges.

I will further add that the same method also suffices to find the distribution of electricity on two parallel, infinitely long, plane strips, whose four edges in cross section form the corners of a rectangle, that is, the cross section of the strips gives two lines which are opposite and parallel to each other. The potential function in this case is given by an equation of the form

where H (u) represents the function designated by Jacobi in the Fundamenta Nova, p. 172, as the numerator of the function developed in terms of sin am u. The overlying strips correspond, according to Jacobi's notation, to the values =±2 K where x=±2 KA gives the half distance of the strips, while the width of the strip depends on the ratio of the constants A and B.

The form of the equations (2) and (4) allows us to recognize that and can be expressed as function of x and y only by means of most complicated serial developments.

IV.

ON A THEOREM RELATIVE TO MOVEMENTS THAT ARE GEOMETRICALLY SIMILAR IN FLUID BODIES, TOGETHER WITH AN APPLICATION TO THE PROBLEM OF STEERING BALLOONS.*

By Prof. H. VON HELMHOLTZ.

The laws of motion of cohesive and non-cohesive fluids [namely, liquids and gases] are sufficiently well known in the form of differential equations, that take into consideration not only the influence of exterior forces acting from a distance, as well as the influence of the pressure of the fluid, but also the influence of the friction [namely, both internal and external frictions, or both viscosity and resistance]. When in the application of these equations one remembers that under certain circumstances [namely, wherever a continuous motion would give a negative pressure] there must form surfaces of separation with discontinuous motion on the two sides, as I have sought to prove in a previous communication to this academy,t then will disappear the contradictions that by neglect of this consideration have hitherto been made to appear to exist between many apparent consequences of the hydro-dynamic equations on the one hand and the observed reality on the other. In fact, so far as I see, there is at present no ground for considering the hydro-dynamic equations as not being the exact expression of the laws controlling the motions of fluids.

Unfortunately it is only for relatively few and specially simple experimental cases that we are able to deduce from these differential equations the corresponding integrals appropriate to the conditions of the given special cases, especially if the nature of the problem is such that the internal friction [viscosity] and the formation of surfaces of discontinuity can not be neglected. The discontinuous surfaces are extremely variable, since they possess a sort of unstable equlibrium, and with every disturbance in the whirl they strive to unroll themselves; this circumstance makes their theoretical treatment very difficult. Thus it happens

* From the Monatsberichte of the Royal Academy of Berlin, June 26, 1873, pp. 501 to 514. Wissenschaftliche Abhandlungen, vol. 11, pp. 158-171, Berlin, 1882.

↑ Berlin Monatsberichte, April 23, 1868. See also No. III of this collection of Translations.