Physical Society, March 9.—Prof. A. W. Rücker, F.R.S., President, in the chair.—Prof. O. Henrici, F.R.S., made a communication on mathematical calculating machines, especially a new harmonic analyser. After mentioning the general principles on which such machines are based, the author showed a new arithmometer devised by Prof. Selling, in which the jerky motions of the numeral wheels common in such instruments are eliminated, and the operations simplified. Another arithmometer of very compact design, named the “Brunsviga,” had been placed on the table by Prof. Boys. The simple and ingenious “hatchet” integrator was then shown. It resembles a small hatchet with a tracing-point projecting at right angles to, and at the end of, the handle. Moving the point from near the centre of mass of any closed curve, round the curve once and back to the starting-point, the distance between the initial and final positions of the hatchet-head is a measure of the area of the curve. The instrument has been found very useful for indicator diagrams. A Hine and Robertson's planimeter (lent by Prof. Perry), an Amsler planimeter combined with a pento-graph for measuring small areas, an Amsler integrator to give areas and first moments, and a beautiful sphere and cylinder rolling-integrator of great accuracy, by Coradi of Zürich, were shown, as well as an ingenious integraph devised by Abdank Abakonowitcz. Passing on to harmonic analysers, Prof. Henrici explained the object of such instruments, viz. to determine the coefficients in Fourier's expansion for any periodic curve, cos 26+X sin 0 + B2 sin and briefly described Lord Kelvin's instrument now in use at the Meteorological Office. This machine gives the first term and three pairs of coefficients A and B, but is large and expensive. The author had endeavoured to devise a simple and more portable instrument, and now described the various stages in the evolution ot his new analyser. Using Clifford's method of wrapping the curve round a cylinder, he saw that by imparting a simple harmonic motion to a plane tangential to the cylinder, which plane carried an Amsler planimeter whose tracing-point followed the intersection of the plane with the curve, as the cylinder rotated, any coefficient An or B"could be determined. This arrangement had considerable friction, and only gave one coefficient at a time; it also necessitated readjustment of the period of the haimonic motion for each pair of terms. Another machine founded on integration by parts was then constructed, in which the relative periods of cylinder and registering wheels was adjusted by a disc and roller, the motion being transmitted to the wheels by bands driven from the disc spindle. This gave A"and ˜Bn at one operation. Mr. A. Sharp used this machine for some time and then designed an inversion of it, in which the curve was laid out flat and the machine rolled over it. This arrangement greatly facilitated the multiplication of registering wheels, and thereby enabled several pairs of coefficients to be determined at once. The first machine of this kind showed several small errors which were avoided in a second instrument, a specimen of which, made by Coradi, was exhibited and described. A rectangular frame carried on three rollers (two being fixed to the ends of a long axis) traverses the paper in the direction of y, and the tracing point is fixed to a carriage which moves on the frame in a direction perpendicular to y, i.e. in the direction of 6. A band is attached to this carriage and imparts a motion proportional to B to two horizontal axes (one for the A coefficients, and one for the B's), placed above and parallel to the long roller axis above mentioned. Each of the two axes carries five pinions having teeth in the ratios I, 2, 3, 4, 5, respectively, which gear with crown wheels fixed to vertical spindles. The latter, therefore, rotate through angles proportional to 0, 20, 30, 40, and 50. To the lower ends of these spindles horizontal rings are attached, in which the bearings of a registering wheel are formed; each wheel rests on a cylinder carried by the long axis, and rolls or slides thereon according as its axis is parallel or perpendicular to that of the cylinder. Moving the tracing-point once round the curve gives five pairs of coefficients. By changing the driving-band to other pulleys so as to turn the pinions at different rates relative to the 0 movement, the 6th, 8th, loth, and 7th and gth pairs can be determined. The chief drawback of the instrument is that the registering wheels are not easy to read, whilst the back-lash of the crown wheels and pinions introduces small errors. In the latest form of instrument toothed wheels are dispensed with, and glass spheres carried in frames on the vertical spindles roll on the horizontal cylinders; each sphere actuates two registering wheels on fixed areas at right angles to each other, and these give respectively the sine and cosine coefficients. The number of vertical spindles is therefore halved, and the instrument greatly simplified. These details have been introduced by Coradi. A working drawing of another analyser, designed by Mr. Sharp, which gives the amplitude and epoch of the curve resulting from each pair of terms in Fourier's expansion, was exhibited. The discussion on Prof. Henrici's communication was postponed until next meeting.-Mr. H. Wilde, F.R. S., then exhibited and described his "magnetarium."This consists of a hollow geographical globe> wound all over the inner surface with insulated wire in planes parallel to the equator. Within this globe is a sphere wound with wire on its surface, and having its axis inclined at 23^° to that of the outer globe. By meins of epicyclic gearing the spheres can be made to rotate at slightly different rates. When electric currents of suitable strength are passed through the two windings, the magnetic condition of the earth can be imitated, both as regards distribution at any epoch, and the secular variations. A better result was obtained by putting sheet iron over the land areas, and a still closer approximation by using thin iron over the water areas. A magnetic chart and tables giving the magnetic elements at various places for different epochs as determined by the magnetarium were shown. The author mentioned that recent observations by the United States Survey at Ascension Island, and by Prof. Thorpe in Senegambia, had confirmed results obtained by his instrument. The President said he had tried the apparatus, and found the Siberian oval closely imitated. The secular variations at Greenwich were also well shown. In South America the approximation was not so good. In reply to a question by Mr. Blakesley, Mr. Wilde said the present position of the pole of the inner sphere was 84° W., 67° N.