The following observations upon ruled surfaces generally may also be quoted:

"It is not possible to express the general equation of ruled surfaces in a purely functional form, for the equations of a right line are only two in number, and they involve four arbitrary constants. We cannot clear all four by such suppositions as we have made in the previous cases without introducing some similar restriction. If, for instance, we take ax = y + b,

in which b remains perfectly arbitrary, and as it is not a function of x and y alone, we cannot get rid of b without bringing back one of the other arbitrary quantities or introducing a restriction. We cannot expunge them without partial differential equations.

In

"It is also to be remarked that the functional equations which we have hitherto obtained are not quite general. the case of cones, for instance, we have given the equation

y

b

=

y F

on the assumption that the vertex is at

the known point, whose coordinates are (a, b, c). If the vertex is to be left arbitrary, so that the functional equation shall express all conical surfaces without reference to the position of the vertex, no such functional equation exists, nor can the condition in general be expressed by any one differential equation.

"We have seen that we cannot express the equation of a ruled surface in a functional form without differentiation.

This happens because the simplest expression of a ruled surface is that its tangent plane at any point shall meet it in a right line, or, what is the same thing, that one of its tangent lines at any point shall be wholly in the surface. Now the question of tangency is emphatically a question of differentiation.

"A ruled surface is expressed with complete generality by considering it to be traced out by the motion of the generating line

in which the four quantities c, which are constant so far as the equation of the line in any one position is concerned, but variable parameters, when considered with reference to the position of the line, are to be made to disappear. The obvious way of effecting this is to obtain, by means of implicit differentiation, a relation between the partial differential coefficients of z, which, when thus cleared of what is special to the particular generating line, will be the differential equation, in partial differentials, of the surface. "Operating implicitly upon the above equations, we get

"Differentiating again upon the same suppositions, we have

"If now, by ordinary algebra, we eliminate c, from equations (2) and (3), we get the differential equation of ruled surfaces. It is not worth while to write it down, as it is more conveniently used in the implicit form given above than in its very cumbrous explicit form.

"We have already noticed that the general form of a ruled surface cannot be expressed as a single functional equation. It follows that the differential equation has no general primitive.'

*a, B, Y,

and d being the partial differential coefficients of the third order.

The least complete part of this little treatise appears to be that in which Mr. Merrifield endeavours to separate the cones and cylinders from other families of surfaces by means of the third differentials. We believe he was the first to remark that, to use his own language:

"The equations of the cones

in virtue of the relation rt=s2, lead to the equation of the third degree

4 (ay — B”) (BS — y3) = (ad — By)2.”

But Dr. J. Hopkinson has shewn that this last equation is true for any developable surface whatever. It does not, therefore, distinguish the cones. In the case of the cylinders, again, when, as Mr. Merrifield shewed, each factor vanishes separately, or, what is the same thing, a, ß, y and 8 are in continued proportion, Prof. Cayley has shewn that the equations can be integrated, and lead to a family of surfaces much more general than the cylinders. Mr. Merrifield suggests that these equations distinguish the cylinders from other developable surfaces. This may be so, but, of course, a proof would be desirable.

Apart from this, which is a mere deficiency, and no actual error, this catalogue is a very useful guide to a most interesting collection, and we would seriously advise all students of Solid Geometry to go to South Kensington, and study the models with the assistance of the catalogue, which they can buy at the door of the Museum.

*Mr. Merrifield had also noticed that it included the ruled surface with a director plane.

NOTE ON HYDRODYNAMICS.

By E. J. Nanson, B.A.

IN a paper on the Integrals of the Hydrodynamical Equations which express Vortex Motion, Helmholtz has shewn that in the case of homogeneous incompressible fluid the vortex lines move with the fluid. This result he derives from three equations, of which the type is

dw

where =

dv

dy-dz, &c., and in forming the differential co

efficient we follow an element of fluid.

dt

In proving these equations use has been made of the equation of continuity, in addition to the equations of equal pressure. Thomson has extended the theorem to any fluid in which the density is a function of the pressure, and in his proof the equation of continuity is not required.

The following is a direct proof of the theorem as extended by Thomson:

With the same notation as above, it may be deduced from the equations of equal pressure that

Let q denote the vector of which the three quantities §, n,

d

are components, and let the symbol denote space-differ

db

entiation along the vector q. Also let Ø denote

Then the above equation may be written

du dv dw

+ + dx dy dz

==

.(2).

These equations shew that the vortex lines move with the

fluid.

If we introduce the equation of continuity, which may be written

This form of the equation is precisely similar to (1),

ξη ζ

-

being used instead of §, n, C. Hence, the reasoning of Helmholtz will apply almost word for word to demonstrate the truth of the theorem that the product of the cross section of a vortex filament into the angular velocity is constant. In Thomson's paper, however, it is shewn that this result is a consequence of the equations of equal pressure only.

It may be noticed that equation (3) verifies at once the first integrals of the equations of motion which are due to Cauchy. They are given in the Cambridge and Dublin Mathematical Journal, Vol. III., p. 212, and may be written

where

denotes the initial value of and a, b, c are the

ρ

initial coordinates of the particle of fluid which at the time t is in the position x, y, z.