mean temperature of the place. With these data, the increase is 1° F. for every 54.57 feet, which approximates to that obtained by Professor Phillips at Monkwearmouth of 1° F. for about every 60 feet.

If, on the other hand, for the purpose of comparison, we adopt the measurements for the invariable stratum as obtained at Dukenfield, we find the rate of increase to be 1° F. for every 47.2 feet as against 1° F. for every 83.2 feet in the case of Dukenfield itself. So great a discordance in the results is remarkable, and is not, in my opinion, attributable to inaccuracy of observation in making the experiments. On the other hand, I may venture to suggest that it is due, at least in some measure, to dissimilarity in the position and inclination of the strata in each case. These I now proceed to point out.

Position of the Strata at Rose Bridge and Dukenfield Collieries.— Rose Bridge Colliery occupies a position in the centre of a gently sloping trough, where the beds are nearly horizontal; they are terminated both on the west and east by large parallel faults which throw up the strata on either side. The colliery is placed in what is known as "the deep belt." Dukenfield Colliery, on the other hand, is planted upon strata which are highly inclined. The beds of sandstone, shale, and coal rise and crop out to the eastward at angles varying from 30° to 35°. Now I think we may assume that strata consisting of sandstones, shales, clays, and coal alternating with each other are capable of conducting heat more rapidly along the planes of bedding than across them, different kinds of rock having, as Mr. Hopkins's experiments show, different couducting-powers. If this be so, we have an evident reason for the dissimilar results in the two cases before us. Assuming a constant supply of heat from the interior of the earth, it could only escape, in the case of Rose Bridge, across the planes of bedding, meeting in its progress upwards the resistance offered by strata of, in each case, varying conducting-powers. On the other hand, in the case of Dukenfield the internal heat could travel along the steeply inclined strata themselves, and ultimately escape along the outcrop of the beds.

I merely offer this as a suggestion explanatory of the results before us, and may be allowed to add that the strata at Monkwearmouth Colliery, the thermometrical observations at which correspond so closely with those obtained at Rose Bridge, are also in a position not much removed from the horizontal, which is some evidence in corroboration of the views here offered.

All holes vertical in solid at bottom of pit drilled with water 1 yard deep, and thermometer remained in hole thirty minutes and made airtight with clay.

II. "On the Action of Rays of high Refrangibility upon Gaseous Matter." By JOHN TYNDALL, LL.D., F.R.S., Professor of Natural Philosophy in the Royal Institution. Received December 4, 1869.

This paper is an expansion of the Rescarches already communicated to the Royal Society on the Chemical Action of Light on Gaseous Matter. (See Proceedings, vol. xvii. p. 92.)

III. "On the Theory of Continuous Beams." By JOHN MORTIMER HEPPEL, Mem. Inst. C.E. Communicated by Prof. W. J. MACQUORN RANKINE. Received December 9, 1869.

(Abstract.)

The chief object of the present communication is to remedy some acknowledged defects in the theory of the above-mentioned subject. The principal steps by which it has reached its present state of development are also noticed, and may be briefly recapitulated as follows:

In 1825 M. Navier investigated the conditions of a straight continuous beam resting on any number of supports. His method, though perfectly correct for the assumed conditions (which embraced most cases occurring

in practice), was so exceedingly intricate when the number of openings became at all large, that in such instances it was of little practical use.

In 1849 M. Clapeyron, a distinguished engineer and savant, devised a much more direct and easy means of treating such cases, though he did not at first succeed in giving to his own method all the simplicity and elegance of which it was capable.

This was first done in 1856 by M. Bertot, civil engineer, who, by effecting an elimination which had escaped Clapeyron, arrived at a remarkable equation which has been the key to all subsequent treatment of the subject. This equation involves the bending moments over any three conse cutive points of support, and is well known in France by the name of the "Theorem of the three Moments."

In 1857 M. Clapeyron himself and M. Bresse, Professeur de Mécanique appliquée à l'Ecole Impériale des Ponts et Chaussées, appear to have discovered this theorem independently of M. Bertot, and M. Bresse shortly afterwards extended it to a much greater degree of generality.

M. Bresse's researches on this subject are published in the third volume of his Cours de Mécanique appliquée,' Paris, 1865; but they had been communicated by him to the Academy of Sciences in 1862, and fully completed in the previous year. M. Bresse not only contributed to the advancement of the theory, but entered largely into the best methods of its application to practice, and framed rules which have since, under an Imperial Commission, acquired the character of legislative enactments.

M. Bélanger, Professeur de Mécanique appliquée à l'École centrale, appears, about the same time as M. Bresse, to have made an independent investigation of this subject, and to have brought the theory of it to about the same stage of advancement.

Little has been since added to this theory in France, but valuable contributions to its development in reference to practice are to be found in the works of MM. Renaudot, Albaret, Molinos et Pronnier, Colignon, and Piarron de Mondesir.

In England Professor Moseley is the first writer on mechanics who appears to have occupied himself with this subject. In his work on 'The Mechanical Principles of Engineering and Architecture,' he gives several examples of the application of M. Navier's method to important practical cases. This work was published in 1843, and no doubt furnished the groundwork for Mr. Pole's more extended investigations.

In 1852 Mr. Pole had to examine the case of the bridge over the Trent at Torksey, involving some new conditions not treated by Moseley, but which he found the means of treating with perfect success. About the same time Mr. Pole had to deal with the much more complex and important case of the Britannia bridge, in which, besides variation of load from one span to another, variation of section also had to be considered, and imperfect continuity over the middle pier. These conditions were successfully imported into this method of Navier, which was, however,

only known to Mr. Pole through the examples of its application given in Moseley's work, and the results obtained were identical with those which would have followed from the application of the method of Clapeyron in its most improved and generalized form.

In 1858, the present writer, being then in India, had occasion to consider the condition of a continuous girder of five spans, and finding the method of Navier unmanageable, was forced to seek for some other. He first came upon the equation which he afterwards found had been for some years known in France as the "Theorem of the three Moments," and afterwards extended it, so as to take in all the conditions of the Britannia bridge and to verify all Mr. Pole's results. In this form it was absolutely identical with the equation given by M. Bélanger, and nearly so with that of M. Bresse.

The great defect in all this theory up to the present time has been that, in order to avoid an inextricable complexity, it has been necessary to consider the load in each span as uniformly distributed over it, and the moment of inertia of the section as uniform throughout each span.

In many cases these hypotheses are false, notably so in the case of the Britannia; and the conclusions are affected by their falsity, to what extent being a matter of uncertainty, though good grounds have been shown for believing that the errors cannot attain to importance.

The method now given treats these conditions, it is hoped, rigorously; and although the equations obtained are such as necessarily require some laborious computation to obtain numerical results, they are certainly by no means inextricable.

It is satisfactory to find that in the case of the Britannia, where these new conditions enter with much greater force than in most cases, their effect on the resulting stresses is very unimportant; so that the inference may legitimately be drawn that in all ordinary cases the method of Bresse may be confidently applied.

It is scarcely possible in a short abstract to give an idea of an analytical investigation. The equations obtained are of the same form as those of the previous methods, each containing, as unknown quantities, the bending moments over three consecutive supports; but the coefficients are somewhat involved functions of the varying loads and sections. An abbreviated functional notation has, wherever possible, been used, by means of which a certain degree of clearness and symmetry is preserved in expressions which would otherwise become inextricably complex.

IV. "Remarks on Mr. Heppel's Theory of Continuous Beams." By W. J. MACQUORN RANKINE, C.E., LL.D., F.R.S. Received December 22, 1869.

(Abstract.)

The author states that the advantages possessed by Mr. Heppel's method will probably cause it to be used both in practice and in scientific study.

With a view to the instruction of students in engineering science, he proposes an abridged way of stating the theoretical principles of Mr. Heppel's method, considering at the same time that Mr. Heppel's more detailed investigation forms the best model for numerical calculation.

He then uses Mr. Heppel's improved form of the "Theorem of the three Moments" to test the accuracy of the formula which he obtained in another way, and published in ‘A Manual of Civil Engineering,' for the case of a uniform continuous beam with an indefinite number of equal spans, the successive spans being loaded alternately with a uniform fixed load only, and with a uniform travelling load in addition to the fixed load; and he finds the results of the two methods to agree in every respect.

V. "Remarks on the recent Eclipse of the Sun as observed in the United States." By J. N. LOCK YER, F.R.S. Received December 7, 1869.

By the kindness of Professors Winlock, Morton, and Newton, I have been favoured with photographs, and as yet unpublished accounts, of the results of the recent total eclipse of the sun observed in America. I am anxious, therefore, to take the opportunity afforded by the subject being under discussion, to lay a few remarks thus early before the Royal Society. The points which I hoped might be more especially elucidated by this eclipse were as follows:

:

1. Is it possible to differentiate between the chromosphere and the corona? 2. What is the real photographic evidence of the structure of the base of the chromosphere in reference to Mr. W. De La Rue's enlarged photographs of the eclipse of 1860?

3. What is the amount of the obliterating effect of the illumination of our atmosphere on the spectrum of the chromosphere?

4. Is there any cooler hydrogen above the prominences?

5. Can the spectroscope settle the nature of the corona during eclipses? With regard to 1, the evidence is conclusive. The chromosphere, including a "radiance," as it has been termed by Dr. Gould (the edge of the radiance as photographed being strangely like the edge of the chromosphere in places viewed with the open slit), is not to be confounded with the corona.

On this subject, in a letter to Professor Morton, Dr. B. A. Gould writes:-"An examination of the beautiful photographs made at Burlington and Ottumwa by the sections of your party in charge of Professors Mayer and Haines, and a comparison of them with my sketches of the corona, have led me to the conviction that the radiance around the moon in the pictures made during totality is not the corona at all, but is actually the image of what Lockyer has called the chromosphere.

"This interesting fact is indicated by many different considerations. The directions of maximum radiance do not coincide with those of the great beams of the corona; they remain constant, while the latter were variable.

VOL. XVIII.