though they do not actually join them; but when the streams increase in quantity, they join and excavate courses for themselves; and these actually run into the main watercourse which bounds the district, and so cut out a river-bed, which, whether full or empty, forms a visible mark on the earth's surface. No such action takes place at a watershed, which therefore generally remains invisible.

There is another difficulty in the application of the mathematical theory, on account of the principal regions of depression being covered with water, so that very little is known about the positions of the singular points from which the lines of watershed must be drawn to the summits of hills near the coast. complete division of the dry land into districts, therefore, requires some knowledge of the form of the bottom of the sea and of lakes.

Ρι

On the Number of Natural Districts.

A

Let p, be the number of single passes, p, that of double passes, and so on. Let b1, bg, &c. be the numbers of single, double, &c. bars. Then the number of summits will be, by what we have proved, S=1+P1+2P2+ &c.,

and the number of bottoms will be

I=1+b2+2b2+ &c.

The number of watersheds will be

W=2(b2+p1)+3(b2+P2) + &c.

The number of watercourses will be the same.

Now, to find the number of faces, we have by Listing's rule

P-L+F-R=0,

where P is the number of points, L that of lines, F that of Faces, and R that of regions, there being in this case no instance of cyclosis or periphraxy. Here R=2, viz. the earth and the surrounding space; hence

F-L-P+2.

If we put L equal to the number of watersheds, and P equal to that of summits, passes, and bars, then F is the number of Dales, which is evidently equal to the number of bottoms.

If we put L for the number of watercourses, and P for the number of passes, bars, and bottoms, then F is the number of Hills, which is evidently equal to the number of summits.

If we put L equal to the whole number of lines, and P equal to the whole number of points, we find that F, the number of natural districts named from a hill and a dale together, is equal to W, the number of watersheds or watercourses, or to the whole number of summits, bottoms, passes, and bars diminished by 2.

LI. On Solar Protuberances (being an extract from a Supplement to his second Paper). By Professor J. C. F. ZÖLLNER*.

AS

MY DEAR SIR,

To Dr. Francis, F.L.S. &c.

2 Thurlow Place, Lower Norwood, London, S.E., November 14, 1870.

S the Government has determined to afford assistance to the expeditions that will proceed to Spain and Sicily to observe the solar eclipse in December next, I send to you without loss of time a translation of a part of a paper by Professor Zöllner,

*From. No. 1772 of the Astronomische Nachrichten.

containing his description of an arrangement whereby the solar protuberances are rendered readily visible by means of a telecope of only 12 inches focal length combined with a spectroscope. Yours faithfully, W. G. LETTSOM.

The magnitude of the solar image in the refractor, or, in other words, the focal length of the object-glass employed, plays a very prominent part in the entire method of observing the protuberances. It follows directly from the theory developed in a former part of this paper, that with one and the same spectroscope the contrast between the protuberance and the general ground is dependent upon the width of the slit alone. But now, as with a constant width of the aperture a so much greater part of the protuberance is seen at once the smaller the image of the sun is, it follows that we should endeavour to obtain the amplification of the protuberances we wish to observe, not by means of the solar image (that is to say, not by the employment of an object-glass of great focal length), but rather, as much as possible, by means of the lenses of the spectroscope; and this can be readily brought about by having recourse to a collimator of short focal length compared with that of the observing-telescope. Assuming, for instance, that we have a refractor of 10 feet focal length to which a spectroscope is adapted, the focal length of both object-glasses being equal-if in this state of things it is necessary to open the jaws of the slit 1 millim. in width to obtain at once a view of the whole of a protuberance of a certain extent, the opening of the slit might be reduced to one-tenth of its former width, provided the image of the sun were ten times as small; while the protuberance would still remain visible to its entire extent, and would be seen in ten times as strong contrast relatively to the ground of the spectrum; while, in order to arrive at the same amount of amplification of the protuberance in the field of view (an amplification that we have sacrificed by a diminution of the size of the sun's image), all we have to do is to give the collimator a focal length ten times as short as that of the spectroscope. We should therefore, adhering always to the instance we have selected, with the same optical amplification of the protuberance and with the same system of prisms, obtain a ten times as good effect by employing, instead of a refractor of 10 feet focal length, an instrument of only 1 foot focal length, and by giving to the collimator a focal length of about 2 inches, and to the observingtelescope a focal length of about 20 inches.

The quality of the images, as far as it is dependent on the system of lenses, would be very little affected thereby, inasmuch

as the imperfections arising from chromatic aberration are altogether absent, owing to the homogeneous quality of the light of the protuberances; and hence, as I have convinced myself by repeated experiment, non-achromatic lenses, when suitably selected, may be employed without hesitation for arrangements of the nature here contemplated.

This extremely compact form of instruments suitable for observing the solar protuberances admits of a delicate motion being given to them with facility by clockwork, and holds out the prospect of our seeing realized by these simple means very shortly the idea already broached by me in my former paper, of obtaining an artificial solar eclipse, of any duration desired, for the simultaneous observation of all the protuberances situated on the edge of the sun.

I

Leipzic, August 26, 1869.

LII. On the Principles of Thermodynamics.

By the Rev. J. M. HEATH*.

HAVE to apologize to Mr. Rankine for attributing to him an admission which it appears he never intended to make. I understood him to mean that the period in which the best and original writers on thermodynamics had been careful observers of the true principles of the science was to be dated from the time when the revived theory of molecular vibration in gases had superseded the older one of centres of repulsive force; and I believed that this event did not happen until long after the first speculations in thermodynamics. But since Mr. Rankine disclaims having made any such admission at all in any shape, I of course acknowledge my mistake and beg to withdraw the assertion. But this point is not very material to the main purpose of the argument of this discussion. The new position from which we now start is this. Mr. Rankine contends that all good writers on thermodynamics, up to the very earliest of them, have been reasoning correctly and from properly assumed premisses; whereas my complaint is that all those same writers, down to the very latest of them, have been and are reasoning from premisses improperly assumed at the beginning. To be of any use, therefore, this discussion must now be turned to the consideration of what these assumptions are, whether or not they have been justifiable, and, if not, what others ought to have been assumed in place of them.

Mr. Rankine has stated it to be the business of the thermo

* Communicated by the Author.

dynamist to distinguish the forces which have done mechanical work into two groups-those which have accelerated molecular motions, and those which have only "stored up energy" in some previously exhausted magazine. The words are mine and not his; but I have stated the proposition in such a form that I think he will still acknowledge it as his own, while on my side I can also assent to it, which I could not do in the form he had given to it. So far, then, I believe there is entire agreement between us. It is in the next step taken towards making this distinction among the forces that I begin to dissent from him and, I fear, from the unanimous opinion of all scientific men. That step is the adoption, as the rule required, under the name of the first law of thermodynamics, of the condition that it is those forces that "do work,” according to a certain technical definition of that term, which are considered as those which "accelerate molecular motion;" and, of course, by necessary consequence, those which do no work are those which produce no motion, but "store up energy.' The counter assertion, which I rely upon being able to sustain, is that those forces only are employed in generating motion which, according to the definition of work, do no work-and that those which do work generate no motion or heat, but do "store up energy."

Before proceeding to substantiate these opinions, it is as well, to prevent misunderstanding, that I should say what I understand that definition of work done to be, against which I am protesting. I understand, then, that work is done, according to this definition, by a force P, when, in acting upon a body m, it meets with a resistance Q equal and opposite to itself, which it overcomes" by driving m in a direction opposite to the action of Q through a space dv without increasing its vis viva. And the measure of the work so done is fQdv, or, for the sake of simplicity, let us say Qov. If I have misstated this definition, I shall be sincerely grateful to Mr. Rankine or any one else who will point out my error, or show me where a better one has been given. It is against this definition only that I contend; and if Mr. Rankine disowns its correctness, I am and have long been fighting against a mere shadow. But I shall proceed to examine it on the supposition that it is a correct description of what is meant by the work done by P.

The general equation of vis viva, simplified as above for our purposes, is this, (P-Q)&v= This gives us the vis viva

mv2

mv2

2

that is generated in m when urged through a space dv 2 by the action of a force P against a resistance Q.

If P is