. . from A' and B', whereby we get 1 AG.BF.A'F.BG R= (196) p. A'G. B'F The equipotential circles with the centres C and D being common to the system due to the source at A and sink at A', and also to that due to the sink at B and source at B', and having equal but opposite potentials when taken as belonging to either of these systems separately, will, in the system due to the four poles, form the two branches of the line of zero potential. The other equipotential lines of the system due to the four poles consist also each of them of two branches, both of which never lie within the same one of the two circles in question. From this it follows that each circle divides the sheet into two equiresisting portions; and consequently the resistance of each of them is 1 AG.BF. A'F. B'G (22) 41. Since half the lines of flow due to each pole lie within the circle drawn through them all, the resistance of the portion of the sheet bounded by this circle is ($ 25) twice the resistance of the unlimited sheet; consequently it is equal to R' or to R" according to the arrangement of the poles, 42. It was pointed out in § 39 that the resistance of the part of the sheet outside the circular flow-line with centre C, to the flow from the source A' to the sink B', is the same as the resistance of the part inside this circle to the flow from A to B. Accordingly the value of equation (20) remains unaltered when A' and BỊ are put for A and B, and vice versá, and at the same B = PAPB The AB circles round the two sources then coincide approximately with the two branches of a single equipotential line; and the same is true for those surrounding the sinks. Similar remarks are applicable to equation (22). Experimental verifications of some of the conclusions here arrived at will be given in Part II. of this communication. * It may be noted that by adding together the values of R' and R" we 1 AG. BF 2 2πκό .log which, written in the simpler form 2 TKO р is the resistance of a circular disk on whose edge the poles A and B are .placed (see equation 13). Errata in No. 326. 398, line 17, for from line read from one line. get LIV. On a new Revolving Polariscope. By William SPOTTISWOODE, M.A., F.R.S.* THI THIS instrument consists of a Nicol's prism or other ordi nary polarizer, and a double-image prism as analyzer. The latter is so cut as to show one image in the centre of the field of view, the other excentric; and the peculiarity of the arrangement consists in giving to the analyzer a rapid motion of rotation. If the speed attains eight or ten revolutions per second, the image will remain persistently upon the retina during an entire revolution, and all the phenomena which are usually seen in succession will appear displayed simultaneously in a circle or ring by the excentric image. The central image will consist of a superposition of the images due to all the successive azimuths of the analyzer, and will consequently appear unchanged in brightness or in colour during the working of the instrument. In particular, if the polarizer and analyzer be used without any interposed plate, the excentric image will as usual be brightest at two positions opposite to each other, say at 0° and 180°, and dark at the two positions 90° and 270°. A rapid revolution of the analyzer will therefore give the appearance of a ring brightest at the two positions first mentioned, and fading into darkness at the two other positions. If a plate of selenite be interposed with its axis at 45° to the original plane of polarization, the two images will present complementary tints at 0° and 180°, and likewise at 90° and 270°; but at the two latter positions the tints will be the reverse of those at the two former. At intermediate positions the tints will be fainter, while 45° and 135° will be positions at which each tint is passing into its complementary, and all colour is lost. In this case, therefore, the ring will appear coloured in opposite quadrants with the same tint, in the intermediate quadrants with the complementary tint. In the intervening parts the tints fade into one another. If a plate of quartz cut perpendicularly to the axis be used instead of the selenite, the entire series of spectral tints will be seen displayed twice over in the ring. The order of the tints in the ring will for a given direction of revolution depend upon the character of the quartz, i. e. whether it is positive or negative. Some interesting experiments may also be made with a quarterundulation plate. By this means plane polarization may, as is well known, te converted into circular, and circular into plane. Hence, if we place a quarter-undulation plate in front of a selenite, we shall produce the complete series of tints in the ring, as with a quartz plate. The order of the tints will be that due to a right-handed quartz for one position of the quarter-undulation plate, and will be that due to a left-handed quartz for a position at right angles to the first. * Communicated by the Author In order that this effect may be successfully produced, the thicknesses of the selenite and quarter-undulation plate must be adapted to one another. The latter is usually constructed for the yellow rays; and with such plates we should consequently use a selenite giving yellow and blue images. If the quarter-undulation plate be used with a quartz perpendicular to the axis, the tints may be reduced to a pair of complementaries, as given by a selenite. To these many other and varied experiments may be added, but on the present occasion I will confine my remarks to two of them. In the cases hitherto described the central image appears colourless and uniformly illuminated, while the excentrie image or ring has been the chief subject of observation. But if instead of a plane plate we use quartz wedges giving Savart's bands, or a concave quartz cut parallel to the axis, the central image will be the most interesting. We shall then see light and dark (or coloured and colourless) bands taking alternate places at each half revolution of the analyzer. The ring shows no very striking feature. If a quarter-undulation plate be used, the bands will be seen to travel across the plate, in one direction when it is placed so as to produce right-handed, or in the other when placed so as to produce left-handed circular polarization. Analogous effects are seen with a concave quartz plate cut parallel to the axis. The central figure then shows rings, which with a quarter-undulation plate expand or contract according to the position of that plate. When, however, the analyzer revolves rapidly, it will be noticed that the rings assume the form of spirals. This is due to the fact that the central image, when produced by a circularly concave plate, is not accurately circular, but elliptical, owing to the unequal refraction of the doubleimage prism in two rectangular directions. To the same cause is attributable the apparent wabbling of the central image, even when the instrument is in perfect adjustment. The principle of the revolving analyzer is applicable alike to a table polariscope for eye-observations and to one constructed for projection. In the table polariscope it is used above the analyzer, a diaphragm being placed immediately over the object on the stage of the instrument; in that for projection it may be placed in the focus of the focusing-lens of the system. LV. Proceedings of Learned Societies. ROYAL SOCIETY. [Continued from p. 326.] June 18, 1874.-Joseph Dalton Hooker, C.B., President, in the Chair. “On the Sun-spot Period and the Rainfall." By J. A. Broun, F.R.S. Having read with much interest Mr. Meldrum's communication to the Royal Society on the apparent simultaneity of excess of rainfall and sun-spot area*, I have waited some confirmation of his conclusions from a more extensive induction. Mr. Hennessey's “ Note" in the Proceedings of the Society for April 1874+ induces me to offer the following views and results to the Royal Society, It is well known that the amount of rainfall is a very variable quantity in some countries and in certain positions, and that when there is a year of drought in one part of the world, there is frequently an excess of rain in another. Any investigation, then, which should be occupied with the average fall of rain over the earth's surface must be long and laborious, unless the variation to be dealt with is large and marked compared with others which must be considered purely accidental relatively to the sun's spots. In proof of this I may cite the rainfall at Mussoorie given by Mr. Hennessey I, where, as far as the sun-spot area is known, any result favourable to the connexion of the two phenomena depends wholly on the rainfall for 1861, which is upwards of 50 inches in excess of the mean. If this excess be not due to the great spot-area, then a long series of years' observations might be requisite to make the positive and negative errors destroy each other. It has been with the intention of determining what may be the effect of a given change of sun-spot area, within a limited district, during a period favourable to the connexion of the two phenomena, that the following discussions have been made. We can then say approximately within what limits the excess and deficiency of rainfall lie for the years of greatest and least spot-area, what amount of observations may be required to destroy accidental variations, and whether the result may encourage more extensive research. Mr. Meldrum finds a mean difference of 8.5 inches of rain between the falls for the years of greatest and least spot-area $ ; but this result is derived to some extent from short series of observations made in different parts of the world, and gives no weight to the rainfall in other years than those considered years of maximum or minimum sun-spots. * Proceedings of the Royal Society, vol. xxi. p. 297. † Ibid. vol. xxii. p. 286. Ei Ibid. vol. xxii. p. 287. $ Ibid. vol. xxi. p. 305, Should there be any connexion betwixt the rainfall and spotarea, we may always in the first instance represent it approximately by an equation of this form, AR=faa, where AR is the excess or deficiency of the rainfall from the mean, AA is the excess or defect of spot-area for the same period of time, and f is a constant to be deduced from the observations. Having obtained the mean spot-area for each year from 1832 to 1867, from Table VII. of the paper on this subject by Messrs. De La Rue, Stewart, and Loewy*, the mean for three periods of 11 years (1832 to 1864) was found equal to 643 millionths of the sun's visible surface ; with this quantity the values of FAA (in millionths of the sun's surface) for each year were obtained. Mr. Meldrum's conclusion_depends chiefly on observations during these periods in Great Britain ; and as he has deduced the rainfall for the first period of minimum spots from observations at three stations, Greenwich, Carbeth (near Glasgow), and Aberdeen, I first examined the observations at these places together with simultaneous observations at Makerstoun for the two periods 1832 to 18537. Applying the above equation to these observations, the following results were obtained : Greenwich ......AR=-0.00092 AA; AR= +0.00158 AA; Aberdeen ..AR= +0.00128 AA. Greenwich and Makerstoun are thus opposed to the conclusion, and Carbeth and Aberdeen are more strongly in its favour. It should be remarked, however, that the result for Aberdeen depends wholly on the rainfall given for that place in 1834 (12:3 in.) being exact. As it is 12 inches less than the mean, while at the other three stations the deficiency is only from 0.6 in. at Greenwich and Makerstoun to 1.2 in. at Carbeth, this may be due to a leaky rain-gauge or to a clerical error of 10 inches. In any case no great weight can be given to the conclusion from these four stations. I now sought for an approximation to the mean fall of rain for Great Britain, and for this end have employed the quantities de * Phil. Trans. 1870, p. 399. † The means for Makerstoun during the years 1832 to 1849 will be found in Trans. Roy. Soc. Edinb. vol. xix. pt. ii. p. 108; the falls for the other years are-1850, 21.49 in.; 1851, 25:57 in.; 1832, 32.20 in.; 1853, 23.54 in. | It may here be noted that the sum of the plus and minus differences of R and the mean rainfall for the four stations during the twenty-two years were Greenwich. Makerstoun. Carbeth. Aberdeen.) Mean fall......... 24:4 in. 26•2 in. 43.6 in. 24.2 in. Sums of AR...... 100.1 in. 67.8 in. 92:4 in. 94:3 in. It will be seen that the sums of differences bave no relation to the mean fall of rain. |