In the following Table III. various experiments of M. Darcy on the volumes of water which flow in given times through pipes of different materials and states of their internal surfaces and different diameters, and under different heads of water, are compared with the above formula.

TABLE III.

New cast-iron pipe, interior clean. Diameter 0.188 metre.

Old cast-iron pipe, interior covered with deposit. Diameter

The velocity v of any film as represented by equation (12), and the efflux QR as represented by equation (17), are given in terms of the velocity o of the central filament, and can only be determined by these formulæ when that velocity is known. The experiments of M. Darcy having made it known in certain cases, they serve to verify the theory as it regards the relation of the efflux per unit of time to the velocity of the central filament. If that velocity could be represented in terms of the head of water and the diameter and length and degree of roughness of the pipe, then the formula by which this was represented being substituted for vo in equation (12) would complete the investigation. I propose to make it the subject of a future com

munication.

XXIII. On a supposed new Integration of Differential Equations of the Second Order. By Professor CAYLEY, F.R.S.*

I

CANNOT assent to the views taken by Professor Challis in his paper in the July Number, "On the Application of a new Integration of Differential Equations of the Second Order to some unsolved Problems in the Calculus of Variations."

In any problem of the calculus of variations, where there are two variables x, y, the condition for a maximum or minimum is obtained in the form

A(dy-pdx)=0;

and if the problem involves no relation between Sa and Sy, Professor Challis says that "we have with equal reason A=0 and Ap=0;" and he goes on to argue that "it cannot be antecedently affirmed that these are identical equations;" and further, that "it is necessary to take account of results deducible from them either separately or conjointly.”

I object to this statement; it seems to me that in order that A(dy-pox) may vanish, the only condition is A=0; we are not concerned with the equation Ap=0, as such, at all. But in certain cases it happens that p is a multiplier of the differential equation A=0; viz. by writing this equation under the form Ap=0 we have an equation integrable per se, and which by its integration gives the integral of the differential equation A=0.

Professor Challis, taking the view that the two equations are distinct from each other, considers (Problem II.) the following question:-"Required the minimum surface generated by the revolution of a line joining two given points in a plane passing

* Communicated by the Author.

through the axis of revolution." Here

A=

1

yq

1+p2 (1+p2)*

The equation Ap=0 is integrable per se; and its integral is

y

√1+p2 =c, or say y=c1+p, being, as is known, the differential equation of a catenary having its directrix coincident with the axis of revolution; in fact the integral is

But there are difficulties as regards the application of this integral to the problem in hand; and Professor Challis is led to consider that the solution of the problem "can be effected only by taking into account an independent integration of A=0;" and this he proceeds to obtain by a method which "consists essentially in first finding, when it is possible, the evolute of the curve or curves of which A=0 is the differential equation, and then employing the involutes thence derivable, which may be regarded as the solution of the equation, to satisfy, either by computation or by graphical construction, the given conditions of the problem." This is perfectly allowable; but after the evolute is obtained, we must take not any involute, but the proper involute of such evolute; we thus have a solution of the differential equation A=0, the same as is obtained by what Professor Challis considers to be the integration of the other equation Ap=0. This seems to me obvious à priori; but I will verify it in regard to the problem in hand. Taking, with Professor Challis, a', y' as the coordinates of that point of the evolute which is the centre of curvature at the point of the involute whose coordinates are x, y, and writing also c=2k (the equation in question is obtained in his paper), the several equations obtained by him are in effect as follows:

(the third of these is his equation (a), taking therein the lower

signs and correcting an accidental or typographical error, viz. is printed instead of). But, from the foregoing equations,

y

2k

which, according to Professor Challis, is the integral, not of the equation A=0, but of the other equation Ap=0.

Edinburgh, August 10, 1871.

XXIV. On the General Circulation and Distribution of the Atmosphere. By Professor J. D. EVERETT, of Queen's College, Belfast*.

AT

T the Meeting of the British Association in Dublin in 1857 a paper was read by Professor James Thomson "On the Grand Currents of Atmospheric Circulation," in which certain new views were propounded, differing greatly from the popularly received theory of Lieutenant Maury †, and accounting for the

* Communicated by the Author, having been read to the British Association at Edinburgh (Section A), August 1871.

†The theory of atmospheric circulation which is adopted by Maury is

prevalent south-west winds in north temperate latitudes in a more satisfactory manner than by Maury's supposed crossing of the upper and lower currents near the Tropic of Cancer.

Three years later, Mr. W. Ferrel, A.M., Assistant upon the American Ephemeris and Nautical Almanack, published (in vols. i. and ii. of the Mathematical Monthly') a series of articles "On the Motions of Fluids and Solids relative to the Earth's Surface, comprising Applications to the Winds and the Currents of the Ocean." Mr. Ferrel begins by referring to a pamphlet which he published on the same subject a few years previously, and concludes by pointing out an important modification in one part of his theory as first published, a modifica tion which he has seen it necessary to make after reading the Report of Professor Thomson's paper in the British Association's Proceedings. Mr. Ferrel's paper, as reprinted from the Mathematical Monthly,' occupies seventy-two pages, of which about sixty are occupied with an elaborate mathematical investigation of the distribution and motion of the atmosphere which would result from the rotation of the earth combined with the heating of the equatorial regions on the hypothesis of no friction. In the latter part of the paper the modifying effects of friction are mentioned, and a theory of general atmospheric circulation, as actually existing, is laid down in such language as to be intelligible to those who are not able or willing to follow the steps of the mathematical investigation.

Without discussing the question of priority, I may say that the views advanced by Mr. Ferrel and by Professor Thomson are substantially the same, as far as they are comparable; but Mr. Ferrel's views are much more fully developed and applied.

As the theory propounded by these two authors (which may appropriately be called the centrifugal theory of atmospheric distribution and circulation) is, I believe, very little known among meteorologists, I think I shall be doing good service in calling attention to it, and in pointing out how some of the numerical quantities involved may be calculated without the aid of the higher analysis employed by Mr. Ferrel.

Since the first draft of the present paper was written, a letter has appeared from Mr. Ferrel in Nature' (July 20), in which he calls attention to some of his principal results, and presents some points rather more clearly than in his earlier publications. that of George Hadley, who published it in the Phil. Traus. for 1735 (vol. xxxix. p. 58). Hadley appears to have been the first to point out the true connexion between the earth's rotation and the easting of the trade-winds. Halley, in 1686 (Phil. Trans. No. 183), had indicated the existence of a circulation of air between the polar and equatorial regions, due to difference of temperature, but he erroneously attributed the easting of the trades to the diurnal wave of heat which runs round the earth from east to west.