214 On the Differences in Lichtenberg's Dust-Figures. [June 13, corresponds very closely with the results of excitation of the fibres in the living animal of the same species (Beevor and Horsley). In all the cases in which there was coarse degeneration in the internal capsule it was, with two exceptions (both hallux cases) grouped on the outer edge of the capsule. Attention is called to the fact that a large proportion of the coarser fibres passing down through the capsule enter the substantia nigra, and these experiments show this tract to be nearly or quite as large as that passing down into the pyramid. These are apparently fibres which have been looked upon as pyramidal, and, as the "pyramidal tract" has been shown to be even more extensive in the medulla and below the decussation than in the internal capsule, it follows that the fibres passing to the substantia nigra are probably replaced by others arising at lower levels. These degenerations show that in the monkey the facial fibres are situated in the middle third of the crus, in which they are mingled with the fibres of the pyramid, and that they do not occupy a space by themselves mesial to the pyramid. VI. "On the Cause of the Differences in Lichtenberg's DustFigures: Preliminary Note." By SILVANUS P. THOMPSON, D.Sc., F.R.S. Received May 9, 1895. As ordinarily produced by dusting a mixture of red-lead and lycopodium upon a surface which has been charged by contact with the knob of a Leyden jar, the dust-figures present a remarkable and hitherto unexplained difference of form. The positive figures consist of white lines branching in stellate or dendritic patterns, whilst the negative figures exhibit red patches of circular or ovate outline. The differences, save in the matter of colour, are not due to the powders used nor to the nature of the dielectric surface chosen for the experiment. They vary only slightly with the nature of the gas; but are more considerably altered by the rarefaction of the air. The author found. that the dendritic patterns of the positive figures are correlated to the brush form of discharge, whilst the rounded patches of the negative figures are due to the silent discharge of electrified winds. When polished metal surfaces are used in air for producing the discharges (as in the usual case when the knob of a Leyden jar is employed), negative electrification more readily discharges itself in a wind, positive electrification less readily, disruptively, as a brush. But where a smooth surface of a peroxide, such as the peroxide of lead, is substituted for a metal knob, positive electrification will discharge itself as a wind, giving rise to white positive figures of rounded outline; while negative electrification will under certain conditions produce a brush discharge from the peroxide surface, giving rise to red dendritic patterns. The author considers these differences to be analogous to the differences observed in the experiments of Oliver Lodge upon the photo-electric loss of charge first observed by Hertz. VII. "Theorems on the Attraction of Ellipsoids for certain Laws of Force other than the Inverse Square." By E. J. ROUTH, F.R.S. Received May 11, 1895. (Abstract.) The object of the author is to find finite expressions for the potentials of an ellipsoidal shell, and of a solid ellipsoid when the law of force is the inverse th power of the distance, being positive or negative. It is shown in the beginning of the paper that the two cases in which is an even integer and an odd integer require different treatment. After discussing some special cases, we come to the first general theorem. Supposing that is even and that the shell is a thin homogeneous homœoid, the potential is found to assume very different forms according as is greater or less than 3, so that the law of the inverse square is just on one side of the boundary. When K >3, the potential can be completely integrated, and an expression is found containing (x-2) terms, and involving only the differentiation of an integral rational function of xyz of к— -4 dimensions. general form at an internal point is The When is <3 the potential takes the form of a single integral where t = (2-). This reduces to the ordinary well-known form when t=0, i.e., when the law is the inverse square. Proceeding next to a thin heterogeneous homoeoid, the density being (§ŋz) where is a function of i dimensions, different cases are found to arise according as K is greater or less than 3 and i greater or less than -2. If >3 and i<«-2 the potential can be completely integrated. A finite expression is given containing not -3 terms, and involving differential coefficients of an For an internal point the general more than κ- where A and A' are two differential operators, and Σ implies summation from h=0 to -4. The potential is also found for an external point. If i>--2 or <3 the potential contains a single integral which reduces to a known form when we can put = 2. There are two standard expressions, one of which is of the form D is a differential operator and R a quadratic function of xyz, which are explained in the paper. Examples are given throughout to illustrate the mode in which these general formulæ are to be used, and full references to all other writings on the same subject as far as they are known to the author. Passing on to a solid ellipsoid the potentials at an internal point for both a homogeneous and a heterogeneous ellipsoid are discussed both when <> and <3 and finite expressions are found, which reduce to known forms when = 2. When the point is external and the strata are similar ellipsoids with any law of density, expressions for the potential are found for the cases = 4, 6, 8, and 10. These are reduced to depend on a single integral. Except when = 4, these can be completely integrated when the solid is homogeneous and in some other cases. When is an odd integer, there is a division of cases according as * is < or >2. In the first of these cases, finite expressions for the potential of a thin homoeoid are found (1) when homogeneous, and (2) when heterogeneous. There are corresponding expressions for a solid ellipsoid. As explained in the text, these results differ in form rather than in substance from some already known, but they are treated in a different manner. In the second case, when >2, the integrations become very long. Finite expressions for the potential of a homogeneous homœoid are, however, found (1) when the force varies as the inverse cube, and (2) K when the force varies as the inverse th power, and > 3, the formulæ for the inverse cube being much simpler than for any greater power. Lastly, the potential of a special elliptic disc is discussed which has the property that the level surfaces are confocals, the force varying as any odd inverse power greater than the square. In this list only those properties have been mentioned which the author believes to be new. Transactions. Presents, June 13, 1895. Berkeley: University of California. University of California Studies. Vol. I. No. 2. 8vo. Berkeley 1894; Bulletin of the Department of Geology. Vol. I. Nos. 5-9. 8vo. Berkeley 1894-95; Report of Work of the Agricultural Experiment Stations. 1892-94. 8vo. Sacramento 1894; A Brief Account of the Lick Observatory, by E. S. Holden. 8vo. Sacramento 1895; [and 7 other University Publications]. 8vo. The University. Berlin :-Gesellschaft für Erdkunde. No. 2. 8vo. Berlin 1895. Zeitschrift. K. Preuss. Akademie der Wissenschaften. Bremen:-Naturwissenschaftlicher Verein. Bd. XXX. Sitzungsberichte. Abhandlungen. Bd. XIII. Heft 2. Bd. XV. Heft 1. 8vo. Bremen 1895. The Society. Cambridge, Mass.:-Museum of Comparative Zoology. Bulletin. Vol. XXV. No. 12. 8vo. Cambridge, Mass. 1895. The Museum. The Academy. Cracow :--Académie des Sciences. Bulletin International. Avril, Entomological Society. Proceedings. 1895. Part 2. 8vo. London. The Society. Transactions (continued). Institution of Civil Engineers. Minutes of Proceedings. Vol. 1895. Royal Photographic Society. Journal and Transactions. Vol. XIX. No. 9. 8vo. London 1895; List of Members. 8vo. London. The Society. Luxemburg-Institut Grand-Ducal (Section des Sciences Naturelles et Mathématiques). Publications. Tome XXIII. 8vo. Luxembourg 1894. The Institute. Munich-K.B. Akademie der Wissenschaften (Math.-phys. Classe). Sitzungsberichte. 1895. Heft 1. 8vo. New York:-American Museum of Natural München. The Academy. History. Bulletin. New York 1895. The Museum. The Academy. New York Academy of Sciences. Annals. Vol. VIII. No. 5. 8vo. New York 1895. Paris:-Faculté des Sciences. [Thèses soutenues pendant l'Année 1894.] 8vo. and 4to. The Faculty. Philadelphia :-Academy of Natural Sciences. Proceedings. 1894. Part 3. 8vo. Philadelphia. American Philosophical Society. Proceedings. No. 143. Vol. XXXIII. The Academy. Vol. XXXII. No. 146. 8vo. The Society. [1893-95]. Pisa-Società Toscana di Scienze Naturali. Atti: Processi Verbali. The Society. Vol. XXVI. The Committee. Sydney-Linnean Society of New South Wales. Proceedings. Vol. IX. Parts 2-4. 8vo. Sydney 1894-95. The Society. Tokyo:-Imperial University. Journal of the College of Science. The Society. Washington:-National Academy of Sciences. Report. 1893–94. 8vo. Washington 1895. Observations and Reports. The Academy. Calcutta :-Meteorological Department, Government of India. Monthly Weather Review. December 1894. 4to. Calcutta ; Meteorological Observations Recorded at Seven Stations in India. December, 1894. 4to. [Calcutta.] The Department. |