NOTICES AND ABSTRACTS OF MISCELLANEOUS COMMUNICATIONS TO THE SECTIONS. MATHEMATICS AND PHYSICS. MATHEMATICS. Introductory Remarks by the President, The EARL OF Rosse. Ir has, I believe, been usual, at least recently, in opening the proceedings, to give, as far as may be practicable, a general outline of the business to be brought before the Section, and some kind of notice of the order in which it is likely to be taken. As, however, many papers are often sent in after the meeting of the Section, and as frequently circumstances arise rendering it necessary to alter the order of proceeding, any notice that can be given must be very imperfect; the daily notices, however, will in some degree supply the deficiency. It has also been usual, I believe, and it is obviously convenient, in some degree to define the general character of the business to be transacted, so that new Members may be enabled better to decide whether to attend this Section or some other. I have made inquiry, and find that already there have been received papers on pure mathematics, applied mathematics, magnetism, light, electricity, and meteorology, besides papers on the construction of philosophical instruments. From the titles of the papers, some idea may be formed of the general character of the business to be transacted; still there are many subjects, in fact several branches of science, which are as yet unrepresented in the papers. First as to the papers on pure mathematics. I need perhaps hardly say that essays on so abstruse a subject cannot be of very much interest except to mathematicians; and even mathematicians, unless the papers happen to relate to the particular branches of mathematics with which they are most conversant, may perhaps be sometimes unable to do more than catch the general scope and leading principles of the paper; still without mathematical knowledge many may often, in the results announced, and indeed in the remarks casually elicited, obtain interesting glimpses into the nature of mathematical processes, and some idea of the progress making in that direction. In applied mathematics there is much more of general interest, and the results are often perfectly intelligible without a special education. I recollect at the Meeting of the British Association at Oxford, the general results of a very abstruse investigation in applied mathematics in physical astronomy, were so brought forward as to rivet the attention of the whole Section. It was an account given in general terms by M. Le Verrier of his researches for the identification of a comet. The discoveries in electricity, magnetism, heat, and light cannot fail to be of great general interest. To the human mind nothing is so fascinating as progress. It is not that which we have long had we most value, but that which we have recently acquired: we especially prize new acquisitions, while we enjoy almost unconsciously gifts of far greater value we have long been in possession of. This is our nature; thus we are constituted; it certainly is not surprising therefore that we should have 1859. 1 a peculiar relish for new discoveries. The interest of a discovery is not usually confined to the discoverer, unless he is very churlish, or even to those who are endeavouring to discover; but it often extends to the whole civilized world. The interest is, however, not lasting; for a time we are dazzled by the brilliancy of the discovery; gradually, however, the impression becomes fainter, and at last it is lost entirely in the splendour of some fresh discovery, which carries with it the charm of novelty. When we reflect upon this, we cannot but perceive how very different the state of the world would have been had mankind from the beginning been in possession of all the knowledge we now have, and there had been no progress ever since. We ask, why have all these wonders been placed before us-hidden, veiled-only to be brought to light by the vigorous use of our faculties? How wonderful from its origin has been the progress of geometrical science! Beginning perhaps 3000 years ago almost from nothing, one simple relation of magnitude suggesting another; the relations becoming gradually more complicated, more interesting, more important, till in our day it expands into a science which enables us to weigh the planets; more wonderful still, to calculate long beforehand the course they will take acted upon by forces continually varying in direction and magnitude. When we ask ourselves such questions as these considerations suggest, and thoughtfully work out the answers as far as possible in their full depth of detail, we become in some degree conscious of the immense moral benefits which the human race has derived, and is deriving, from the gradual progress of knowledge. The discoveries, however, in physical science are often immediately applicable to practice, giving man new powers, enabling him better to supply his many wants. We therefore, who are all, in some degree at least, utilitarians, on that account very naturally regard them with deep interest. I am sure the mere mention of the subject has already suggested to you many of the extraordinary discoveries of latter times; for instance, the production of force almost without limit by heat, and its application to locomotion by sea and land,—the transmission of thought, not slowly by letter, not to short distances by sound, but instantaneously to immense distances by electricity; and when we look around us and see how man has appropriated to his use the properties of light and heat, the powers of wind and water, the materials which have been placed before him in endless variety on the surface of the globe which he inhabits,-that he has effected all this by knowledge accumulated by what we call Science, it is surely not surprising that we should look upon new discoveries with surpassing interest. The mere utilitarian, however, has been often reminded that discoveries the most important, the most fruitful in practical results, have frequently in the beginning been apparently the most barren, and therefore that the discoveries in abstract science are not without interest even for him. I confess, however, that the gradual development of scientific discovery,-in fact, in other words, the steady flow of knowledge into the world-which like a stream becoming broader and deeper as it proceeds points to its own source, to its own origin, which is the origin of man,-I confess that this arrangement appears to me to serve far nobler purposes than merely to minister to the corporeal wants of man, as they increase, or are supposed to increase, with the progress of civilization. What those purposes are, to some extent, I think we may clearly see, though to fathom the full depth of such an inquiry would be beyond our powers. Looking merely on the surface, we perceive that the continual springing up of new facts, new discoveries, in endless succession, the rewards of industry, must tend to make man industrious. It inspires him with hope, entices him to labour with his mindthe hardest of all labour; it quickens his faculties, it forces him to look behind and before, to the past and future, and it promotes in him a high moral training by the influence it exercises over his habits and thoughts. Many, no doubt, will feel anxious to see principles immediately applied to practice; in common language, to see principles made useful: they will be highly gratified in the Mechanical Section. Here they may, perhaps, occasionally see the same thing; but more frequently they will find that the results are but stepping-stones which prepare the way for further progress. These few remarks, which I have made principally for the convenience of new Members, will, I think, be sufficient to give some little idea of the kind of business to be transacted here, and I will not allude to the actual practical results which have immediately followed from the labours of this Section. They have been detailed, and recently, especially by my friend on my right hand, Dr. Robinson; and I will only further add, that I feel much gratified to find so large an attendance of eminent men of science here, ready to correct oversights and supply deficiencies. They, I am well aware, are far more competent to preside here than I can be ; but, with their assistance, the duty will be light; and as the Council, no doubt on good grounds, have made the present arrangement, I will, without hesitation or misgiving, at once proceed with the business. On the Probability of Uniformity in Statistical Tables. By R. CAMPbell. The object of this paper is to find a test for ascertaining whether an observed degree of uniformity or the reverse in statistical returns is to be considered remarkable. Suppose the population to consist of ʼn persons, which we will suppose nearly constant. Let a be the number of years during which observations are taken, and suppose the whole number of phenomena of a certain class occurring to, or presented by, the individuals in the population to be ab. We will suppose the phenomena of a kind which are not likely to occur to the same person more than once in the same year. [Of this nature are most important facts, of which such Tables are formed.] Now suppose we know nothing of the laws by which these facts occur, except that above given, namely, that the total number in a years is ab. Let us see what kind of uniformity (starting from that fact alone) we should expect the Tables to present. Let the people alive in a particular year be A1, Ag... An. of the phenomenon being presented in that year by A1 is it will be presented by A, and not by A, will be n b n an The probability then The probability that Hence we can find the probability of its being presented by A1 only; the probability of its being presented by one person only; the probability of its being presented by two persons only, and so on. These expressions will be very complicated. That for the probability of the phenomenon being presented by b persons only will be (n−1) (n−2)... (n−b+1) ab-1 ab-2 ab-b+1 (1 - ab—b) 2.3.4 .b ab-b b. an-1 - (8 + 1)) . . . . . . an an— 2 an−b+1 ab-b an-(n-i) an Now though these expressions are complicated, we get some very simple results. The above expression would be deducible from the expression for the probability of the phenomenon being presented by b-1 persons only, by multiplying by a factor b(a−2)+n+1 (n-b) (a-b)b• which reduces itself to 1+ It would be deduced from the expression for the probability of b+1 persons presenting it by multiplying by a factor, which reduces to Now these are always greater than 1. This shows that the average number is the most probable one to occur in a particular year. The ratios of the probabilities of b occurring, to that of (b-1), (b−2), &c. are b to b-1, |