The Hounslow Heath base was measured with Ramsden's 100 ft. steel chains, and only differs 0.173 ft., or about 2 inches, from its computed length from the mean base. The Belhelvie base in Aberdeenshire, also measured with the steel chains, differs only 0.24 ft., or less than 3 inches, from the computed length. The difference between the measured and computed length of the Misterton Carr base, near Doncaster, also measured with the steel chains, is only 0·157 ft., or less than 2 inches ; and it will be observed that the difference between the computed and measured lengths of these three bases (measured with chains) is not greater than the difference between the measured and computed length of the Lough Foyle and Salisbury Plain bases (measured with the compensation bars), from which it may be inferred, that bases measured with steel chains are deserving of the greatest confidence ; and when the great simplicity, portability, and cheapness of the chains are compared with the complex, heavy and expensive apparatus of the compensation bars, it may be anticipated that they will be more generally employed than they have been of late years, especially in the colonies, and in countries where the transport of heavy articles is effected with difficulty. The length of the base on Rhuddlan Marsh in North Wales, which was measured with steel chains, differs 1.596 ft. from the computed length ; but from the circumstance that the extremities of the base are very badly situated with reference to the surrounding Trigonometrical stations, the angles being very acute and not well observed, little confidence has been placed in the result of the comparison of its computed and measured length. One of the first practical results arising from the completion of the triangulation is, that it is now possible to engrave the latitude and longitude on the marginal lines of the old sheets of the one-inch Map of England, and this is now being done. The following account of the Trigonometrical operations and calculations has been drawn up by Captain Alexander R. Clarke, R.E. ; this account may be considered an al ridgement of that more detailed account which is now in the press, and will be shortly published. It will be seen that the equatorial diameter of the earth, as derived from the Ordnance Survey, is 7926.610 miles, or about one mile 1 99 20 : greater than it is given by the Astronomer Royal in his 'Figure of the Earth,' and that the ellipticity is 209,33, or as the Astronomer Royal conjectured, something “ greater than 300;" which he gives in , the same paper. The mean specific gravity of the earth, as derived from the observations at Arthur's Seat, was stated in a former paper to be 5.14 ; the calculations have since been revised, and it is now found to be 5.316. The mean specific gravity of the earth, as derived from the only other observations on the attraction of mountain masses on which any reliance has been placed, viz. the Schehallien observations, is, as finally corrected by Hutton, %, or almost 5.0. From the experiments with balls we have the following results : By Cavendish, as corrected by Baily.... 5.448 5.67 5.44 From the pendulum experiments, at a great depth and on the surface, the Astronomer Royal obtained 6.566. Two copies of the new National Standard Yard have recently been received through the Astronomer Royal, and it is obviously necessary that the geodetic measures should be given in reference to the standard ; but not knowing from what scale the standard has been taken, Col. James is unable to say at present in what way the reduction is to be made ; that is, whether by reference to the comparison of the old standards which have been already made, or by the mechanical process of a direct comparison of the Ordnance Standard with the new National Standard. This introductory explanation by Col. James is followed by an account of the Trigonometrical operations and calculations; the following is a brief statement of the results: “ 1st. The four bases of verification, when their measured lengths are compared with their lengths as calculated from a mean of the Lough Foyle and Salisbury Plain bases, show the following discrepancies : Hounslow. Misterton Carr. Rhuddlan Marsh. Belhelvie. +1.596 +0.240 a “2nd. The elements of the spheroid most nearly representing the surface of Great Britain are Feet 0. Miles. Equatorial semidiameter=20926249=3963.305 Polar semidiameter =20856337= 3950-06} compression 1 299.33 “3rd. The elements of the spheroid most nearly representing the whole of the measured arcs considered in this paper are Feet 0. Miles. Equatorial semidiameter=20924969 = 3963.064 1 Polar semidiameter 297.96 =20854743 = 3949:966 } compression “ 4th. The lengths of the degrees of latitude and longitude in Great Britain are as in the following table : Mean latitude. Length in ft. of Length in ft. of Length in st. of Length in ft. of May 22, 1836. The LORD WROTTESLEY, President, in the Chair. The following communications were read : I. “On the Application of Photography to the physiognomic and mental phenomena of Insanity.” By Hugh W. DIAMOND, M.D. Communicated by Admiral Smyth, For. Sec. R.S. Received April 23, 1856. (Abstract.) The position of the author, as Medical Superintendent of the Surrey Lunatic Asylum, has enabled him to make the peculiar application of Photography, of which he gives an account in the present communication. He points out the advantages to be derived from photographic portraits of the insane, as faithfully representing the features of the disease in its different forms, or its successive phases in the same patient, and as affording unerring records for study and comparison by the physician and psychologist. In the course of the paper frequent reference is made to the series of photographic portraits of lunatic patients with which it was accompanied. II. “On the Problem of Three Bodies.” By the Rev.J. CHALLIS, M.A., F.R.S., F.R.A.S., Plumian Professor of Astronomy and Experimental Philosophy in the University of Cambridge. Received May 15, 1856. (Abstract.) The object of the author is to give an approximate solution of the Problem of Three Bodies, equally applicable to the motion of the moon and to that of a planet, in which the forms of the developments of the radius-vector, longitude, and latitude in terms of the VOL. VIII. M time, are directly determined by the analysis. The solution to the first power of the disturbing force is effected by means of the following three equations, in which the letters have the significations usually given to them in the planetary theory : dra h 2μ dᎡ dR dr + dt dt? p? do dr dts dt After substituting in the right-hand side of the first equation, the values of r and 8 given by a first approximation in which the disturbing force is neglected, that side becomes a known function of t. The equation can then be integrated approximately so as to give the development of r in terms of t to the first power of the disturbing force, and to any power of the eccentricity it may be thought proper to retain. By substituting in that term of the second equation which does not contain the disturbing force the value of r thus obtained, the integration of the equation gives the development of in terms of t, and lastly by substitution in the third equation z is similarly developed. The author has shown the practicability of this method by obtaining values of r and 0 to terms of the order of the eccentricity multiplied by the disturbing force. The development of the latitude, and a more particular application of the method to the motion of the moon, are reserved for future consideration. The particular advantages of this mode of solution are, that being free from all assumption as to the forms of the developments, it gives those which are alone appropriate to the problem, and it evolves both the periodic and the secular inequalities by the same process. Terms containing ent as a factor, which are met with in other solutions of the same problem, do not occur in this method; but there are terms containing the factor e'nt, which are shown to be convertible into periodic functions, and to have reference to secular variations of the eccentricity and of the motion of the apse. The paper concludes with some general remarks on the principle of this approximate solution of the problem of three bodies, and an explanation of the analytical circumstances which make it, in common with the |