her shades, her selected company of votaries" (Channing); and that many who are engaged in other occupations are at the same time diligent cultivators of science, and often come forward to add contributions to its stores. Still, it is to those who set themselves apart as men of science or as literary men that the world looks for the steady pursuit of knowledge, and for the guidance of others to the solution of those questions which from time to time press upon the interest or affect the welfare of the community. And where Societies of such are formed men look to them to intermeddle with all wisdom, and leave no part of the wide field of knowledge unexplored.* On the value and utility of such Societies it would be idle in me to expatiate in such an assembly as this. Who knows not that by intercourse with others men have their faculties sharpened, are helped better to understand themselves, and to bring to precision and definiteness their own cogitations, as well as stimulated to explore new fields of inquiry and guided to make new discoveries? It is long ago since Homer said— "By mutual confidence and mutual aid Great deeds are done and great discov'ries made. and ample experience has showed that this holds true no less of those who go in quest of truth than of those who engage in military adventure; for as Plato, after referring to this passage in Homer, says, "in society we all are somehow more alert in deed and word and thought;" and Aristotle, also referring to this pas sage, says that by society "those engaged in great undertakings are rendered more potent to think and to act."§ I content myself with congratulating the Society on its past achievements and its present flourishing condition; and expressing my confidence that the energy which has characterised the members in the past will be no less displayed, and with equally satisfactory results, during the session on which we now enter. * Naturæ rerum vis atque majestas in omnibus momentis fide caret si quis modo partes ejus ac non totam complectitur animo.-Plin. Hist. Nat. vii. 1. + Iliad, x. 265, in Pope's version. Protag., p. 348, D. Nicom. Eth., viii. 1. We need have no misgivings as to the worthiness of the pursuits to which this Society is devoted. To search after truth, to strive, the minister and interpreter of Nature," not only to discover all her facts but to elicit their meaning, to educe the principles which underlie them, and to arrive at the apprehension of the laws which they obey,— "Und was in schwankenden Erscheinung schwebt is an occupation than which none can be more noble, more worthy of the faculties with which we have been endowed, or more pleasing to Him by whom these faculties have been conferred,—for He Himself is Truth; and in proportion as men with sincerity and earnestness seek after truth, in that proportion do they put on the similitude of God and come into sympathy with Him. The visible universe, moreover, is as Goethe has expressed it, "Der Gottheit lebendiger Kleid" (the living mantle of the Deity); and by it He, who is himself invisible, reveals himself to us, making known to us, as the apostle tells us, by the things that are made his eternal power and Godhead (Rom. i. 20). Philosophy, if it follow its normal tendency, leads up to God; for the progress of all true philosophic thought is from the many to the one, from facts to principles, from the relative to the absolute, from phenomena to essence; and it is illegitimately arrested in its proper course, and defrauded of its proper issue, if it be stopped short of the Supreme Essence in whose infinite mind. are the archeal types of all existences-" the forms eternal of created things." Nor is the value of literary and scientific study as a moral discipline to be overlooked. To those engaged in such pursuits there arises an influence which, like some subtle essence, pervades their whole inner nature, and unconsciously, perhaps, to themselves elevates, purifies, and refines it. The study of philosophy, of literature, and of science thus becomes a great moral therapeutic, an instrument of spiritual culture:-" Animum format et fabricat, vitam disponit, actiones regit, agenda et omittenda demonstrat, sedit ad gubernacubum et per ancipitia fluctuentium dirigit cursum."+ "The true philosopher (I use the words of Sir Humphry Davy) sees good in all the diversified forms of the external world. Whilst Goethe, Faust, Prol. + Seneca, Ep. xvi. he investigates the operations of infinite power guided by infinite wisdom all low prejudices, all mean superstitions disappear from his mind. He sees man an atom amidst atoms fixed upon a point in space, and yet modifying the laws that are around him by understanding them; and gaining, as it were, a kind of dominion over time and an empire in material space, and creating on a scale infinitely small a power seeming a sort of shadow or reflection of a creative energy, and which entitles him to the distinction of being made in the image of God and animated by a spark of the Divine mind.” * The following statement, in regard to the number of the present Fellows of the Society, has been drawn up by the Secretary:1. Honorary Fellows: Royal Personage His Royal Highness the Prince of Wales, British Subjects John Couch Adams, Esq., Cambridge; Sir George Foreign Claude Bernard, Paris; Robert Wilhelm Bunsen, Carry forward, VOL. IX. 21 Brought forward, Dumas, Paris; Charles Dupin, Paris; Elias Fries, The following are the Honorary Fellows deceased during the year: British-Sir William E. Logan. Foreign-Adolphe-Theodore Brongniart, Christian 2. Non-Resident Fellow under the Old Laws: 21 34 Sir Richard Griffiths, Total Honorary and Non-Resident Fellows at 4th 3. Ordinary Fellows: Ordinary Fellows at November 1875, 1 56 358 New Fellows, 1875-76.-Rev. Francis Edward Belcombe; 15 Brought forward, 9 Total number of Ordinary Fellows at November 1876, . Total Honorary and Ordinary Fellows at commencement 373 10 363 56 419 Monday, 18th December 1876. SIR WILLIAM THOMSON, President, in the Chair. The following Communications were read: 1. On the Roots of the Equation V ppp = 0. By Gustav Plarr. Communicated by Professor Tait. 2. Applications of the Theorem that Two Closed Plane Curves intersect an even number of times. By Prof. Tait. (Abstract.) The theorem itself may be considered obvious, and is easily applied, as I showed at the late meeting of the British Association, to prove that in passing from any one double point of a plane closed curve continuously along the curve to the same point again, an even number of intersections must be passed through. Hence, if we suppose the curve to be constructed of cord or wire, and restrict the crossings to double points, we may arrange them throughout so that, in following the wire continuously, it goes alternately over and under each branch it meets. When this is done it is obviously as completely knotted as its scheme (defined below) will admit of, and except in a special class of cases cannot have the number of crossings reduced by any possible deformation. The excepted class is that in |