On Saturday, July 27, by a mistake, the hole was bored in the duns 20 feet behind the coal. This point had been exposed for a week or two, and the temperature indicated, 69-2, is therefore rejected by Mr. Wethered.

On Saturday, August 3, another hole was bored in the coal, and gave on Monday the same temperature, 68.5, which had been observed in the first hole in the duns. Another hole in the duns gave, a week later, the same temperature, 68.5. Mr. Wethered adopts this as the true temperature at the depth in question (1367 feet).

The thermometer was now removed to the Speedwell pit, the shaft of which is distant about half a mile from the Deep pit, and observations were commenced in a cross-measure branch, which shortly afterwards cut the Two-feet seam of coal.

On Saturday, August 17, the thermometer was inserted in a hard arenaceous stone, and on Monday the temperature 697 was read, depth 1439 feet. This reading was confirmed from the following Saturday to Monday.

On Saturday, October 12, the Two-feet seam of coal having been cut, the thermometer was inserted in it, and on Monday gave a temperature of 69-7, the same as in the stone further back in the branch in August. The depth was the same within 2 feet.

On Saturday, October 26, a hole was bored in the duns under the Two-feet coal, which again gave 69.7.

The next observation was made in the Great Seam coal of the Speedwell pit, in an advanced level head, opening out new ground, depth 1232 feet. The thermometer was placed in a hole bored in the coal on Saturday, November 2, and on Monday the temperature was 66.7. The same reading was obtained the following week in the duns under the coal.

This was the last of the observations deemed reliable. Two other observations were made, the first in ground from under which coal had been worked, and the second in strata disturbed by faults, but in neither case could reliable results be obtained.

The following is a summary of the temperatures, arranged in order of depth, omitting those which are doubtful.

Comparing each depth with the next, we have the following results:

a remarkably regular progression, especially for observations taken in different parts of a colliery. Comparing the surface with the lowest depth, we have an increase of 26°0 in 1769 feet, which is at the rate of 1o in 68 feet; and comparing the depth of 402 feet with the lowest depth, we have an increase of 20°.0 in 1367 feet, which is at the rate of 1°.0 for 68.35 feet.

The observations appear to have been taken in very favourable circumstances, and with much care and judgment. Being the only observations yet furnished to the Committee from the West of England, they form a very valuable contribution to our knowledge.

Mr. Symons has continued his observations at the depth of 1000 feet in the Kentish Town Well (see Report for 1876, p. 209). During 1877 little was done except to continue the record of the temperature of the well-room, have the roof repaired, and make experiments with respect to the elongation of wires of various kinds. In accordance with a suggestion of Sir William Thomson, a new copper wire, No. 22, was purchased, and the Phillips's maximum thermometer, No. 14,608, of which each degree Fahr. is 0-4 inch in length, was lowered to 1000 feet on January 10, 1878. The first noticeable feature, and a very unsatisfactory one, was, that on March 5, 1878, a little mud was found in the protecting case. It will be remembered that the tube was originally 1302 feet deep, but that on the first attempt to lower the thermometer to 1100 feet in May, 1868, the cord was found to become slack at depths varying from 1070 to 1085 feet. It seems probable that the mud has now risen to 1000 feet. Its extreme softness has been illustrated more than once by the fall of thermometers into it, sometimes from a great height. They have never been broken, nor even had their indices displaced. The new wire stretched more than the old one, but after the first two months the elongation was remarkably uniform. The thermometer having been many years in use, it was thought desirable to reverify it, and on September 20, 1878, it was sent to Kew Observatory for this purpose. Another thermometer was temporarily substituted for it, which was only divided to whole degrees and was read by estimation to tenths. With this thermometer the following observations were taken :—

The wire broke on January 2, 1879, and up to the present time no serious attempt has been made to recover the thermometer, but this has arisen rather from want of leisure than from any difficulty in the operation.

The results given in the following table (which goes back to the beginning of the observations), have all been obtained with one and the same thermometer.

The index error of the thermometer has been determined several times, as follows:

:

The gradual rise of zero here indicated is in accordance with usual experience; and the index errors at intermediate dates have been derived from these by graphical interpolation, that is by drawing a curve in which horizontal distance represents time and vertical distance amount of index error, the curve being drawn so as to pass through the four points determined by the above observations, and being made as smooth as possible. The stretching of the wire is determined by the readings of the

recording apparatus described in the 1869 Report, and the correction to reduce from the actual depth to the depth of 1000 feet is made by allowing 018 of a degree per foot, this being the mean rate of increase found by observation (see Report for 1871). The above table shows that the entire range of the corrected temperatures at 1000 feet is less than half a degree, and that the departure from the mean exceeds a tenth of a degree on only seven occasions out of twenty-nine. Mr. Symons has directed close attention to those readings which differ most from the

mean, but has not yet been able to explain the circumstances on which they depend. The maximum elongation of the wire has been 17 feet, and this gives a correction of 31 of a degree. The gradual accumulation of mud at the bottom would account for a gradual change of temperature always in the same direction, if such had occurred (which is not the case), but will not account for alternations of rise and fall such as the table exhibits.

One of the Committee's slow-acting thermometers has been supplied (at his own expense) to Professor John A. Church, of the Ohio State

University, Columbus, Ohio, to be used for observing the temperature at every 100 feet of depth during the sinking of a shaft, probably to the depth of about 4500 feet, in one of the mines of the Comstock lode in Nevada. Two others have been supplied on similar terms to the Meteorological Office.

With the view of carrying out the resolution, expressed in last year's Report, to commence thermo-electric observations in filled bores, the Secretary has procured from Messrs. Siemens 500 feet of No. 20 copper wire, and the same length of No. 19 soft charcoal iron wire, both of them well insulated with gutta-percha, and has conducted some thermo-electric experiments with them in the Laboratory; but the apparatus is not yet ready for actual use.

Mr. Lebour has improved the form of plug devised by him (on the umbrella principle) mentioned in previous Reports (1876, p. 209, and 1877, p. 199). The apparatus now requires only one wire, and remains collapsed so long as the wire is taut, but opens out and plugs the hole when it becomes slack.

The following corrections are to be made in last year's Report.

In the acount of Dr. Stapff's thermometers, 'steel cap' and 'steel jacket,' should be 'brass cap' and 'brass jacket.'

At a later place in the extract from Dr. Stapff's paper, wet boreholes with standing water,' should be wet bore-holes with running water.'

In the references to papers, in foot-note, 1878, 1876, and 1877,' should be 1875, 1876, and 1877.'

Report of the Committee, consisting of Professor CAYLEY, F.R.S., Professor G. G. STOKES, F.R.S., Professor H. J. S. SMITH, F.R.S., Professor Sir WILLIAM THOMSON, F.R.S., Mr. JAMES GLAISHER, F.R.S., and Mr. J. W. L. GLAISHER, F.R.S. (Secretary), on Mathematical Tables. Drawn up by Mr. J. W. L. GLAISHER.

[PLATE I.]

THE present report consists of two parts: I. An account of the state of the calculation of the factor tables for the fourth, fifth, and sixth millions, with some results of the enumeration of the primes in the fourth million; and II. Tables of the Legendrian functions, with an introduction.

I.-State of the Calculation of the Factor Tables for the Fourth, Fifth, and Sixth Millions.

During the year the calculation has been carried on without intermission under the direction of Mr. James Glaisher. At present the factor table for the fourth million is printed and stereotyped, and will be published immediately; the manuscript of the fifth million is complete and ready for the printer. In the sixth million all the entries by sieves