K D ANY TWO PLACES ON THE SURFACE OF THE EARTH, ETC. tween the meridians passing through the places; the third side of the triangle being that which the problem requires us to determine-namely, the direct distance between the places proposed, and subtending the angle at the pole which is formed by the sum or difference of longitudes. In all cases where the spheric projections are not clearly understood, the development of the above-mentioned pyramids on a plane is the most readily comprehended, and from such a development the rules of calculation for the several cases into which the problem divides itself are deduced with the greatest ease and facility; this, therefore, is the method which we intend to employ in the present instance, and for this purpose, let the straight line CA be drawn in any direction and to any distance at pleasure; and in it assume the point C as the centre of the sphere or vertex of the triangular pyramid; then tan a B tan c fan a A at the point C, make the angles I C B and B C G respectively equal to the given co-latitudes. From the point A, any how assumed in CA, let fall the perpendicular A B, which produce to meet the extension of C G in F. At the point B, make the angle F B H equal to the given sum or difference of longitude, and on B as a centre, with the distance B A as radius, describe the circular arc A PB meeting B H in the point H, and draw HF. Then upon the straight line C F, 5 which is given, construct the triangle FCD, of which the sides CD and FD are respectively equal to C A and F H ; then is D C F the angle at the centre of the sphere, which is measured by the arc GK, the required distance between the places. That the preceding is the true development of the triangular pyramid abstracted from the sphere as already described, will be most clearly understood from its reconstruction. Thus, let the figure A C D F H B, be drawn on pasteboard, and correctly cut out from the card along the several bounding lines A C, CD, D F, F H, H B and B A; then let the figure thus separated be folded up upon the lines C B, C F, and F B, and it will be found that the points A, D, and H coincide, as also do the lines A C, D C, and A B, H B; thus constituting the triangular pyramid developed by the preceding process; the circular arcs IB, B G and G K being the co-latitudes and distances between the places respectively. Let C B, the common intersection of the planes A C B and F C B, be made the radius; then will C A and C F be the secants of the angles B C A and BCF to that radius, and A B,, B F the tangents; and if the radius C B be assumed equal to unity, the several lines just referred to will be represented simply by the trigonometrical quantities or angular functions named on the respective lines; the angles B C A, B C F and DCF being indicated by the small Italian letters a, c, b, placed therein, while the angle F BH is denoted by the Roman capital B. Now, the solution of the problem requires the determination of the angle F CD-b, in the plane triangle CFD; but by examining the conditions, we find that the two sides CD and C F, containing the required angle, only are given, and these of themselves are insufficient to determine the angle; but by the principles of construction, the side FD, which subtends the required angle, is equal to FH in the plane triangle F B H, and this we can readily find from the data, which are the two sides B F, B H and the contained angle F BH-B. By the principles of plane trigonometry, it is FH = √B F2 + BH2 + 2B F. BH cos. FB H; that is, FH = √tan.2a + tan.2 c±2 tan. a tan. c. cos. B. Here, then, we have the side F H expressed in known terms, but F D is equal to FH; hence it follows, that in the plane triangle F C D, all the sides are given to determine the angle subcos. b. = sec.2a + sec.2c-tan.2 a-tan2 c 2 tan. a tan. e cos. B but by the property of the circle in re- ; that is, 17tan. a tan. c cos. B sec. a sec.c we get cos. b; from which last form by the theory of angular sections, we finally obtain cos. bcos. B sin. a sin. c cos. a cos. c. This, then, is the final form of the equation, for calculating the direct distance between any two places on the surface of the earth, where it must be observed, that a is the complement of one látitude, and c that of the other, and that the minus or plus sign is to be employed, according as the angle B, or the side a or c is greater or less than ninety degrees, the cosines of angles between 90° and 180° being negative. The same formula, it is manifest, will apply to the determination of the distance between the sun and moon, or between the moon and a fixed star, when their right ascensions and declinations are known; so that this problem, when taken in connexion with that at page 410, vol. xlii., constitutes one of the best methods of finding the longitude of a ship at sea, and that which is generally known to seamen by the name of Lunar Observations. Our present object, however, is only to determine the distance between places on the earth, and for this purpose, we must consider the problem under the various cases into which it naturally divides itself, and these are, 1. When the latitudes are both north or both south, and the sum or difference of the longitudes less than ninety degrees or a right angle. 2. When the latitudes are both north or both south, and the sum or difference Longitude of Halifax 63° 37' 30" West. Longitude of Liverpool 2 59 30 West. Diff. of longitudes...... 60 38 0 ... 44 39 24 Latitude of Liverpool 53 24 36 of the longitudes greater than ninety degrees or a right angle. 3. When the latitudes are, the one north and the other south, and the sum or difference of the longitudes less than ninety degrees or a right angle. 4. When the latitudes are, the one north and the other south, and the sum or differences of the longitudes greater than ninety degrees or a right angle. These are the four distinct cases of the problem, and the method of applying the above general formula to the resolution of each case, will become manifest from what follows. EXAMPLE 1. What is the direct distance in English miles, between St. Paul's Church at Liverpool, and the pillar in the Dockyard at Halifax, North America, the latitudes being respectively 53° 24′36′′ and 44° 39′ 24′′ north, and the longitudes 2° 59′ 30′′ and 63° 37′ 30′′ west of Greenwich? Here the latitudes are both north and the longitudes both west, which corresponds to the first of the preceding cases, and also to that represented by the construction; for since the longitudes are both of one name, the angle contained between the meridians of the two places is equal to the difference of the longitudes, and being less than ninety degrees, the conditions of the first case are completely satisfied. The operation is therefore as follows: Log. cos. 9.690548 Log. sin. 9.846867 Log. sin. 9.904673 Log. sin. 9.751540 Nat. num. +0.56434 therefore we have b=50° 33′ 30"-3033.5 geographical miles; but the geogra ANY TWO PLACES ON THE SURFACE OF THE EARTH, ETC. phical mile is to the English mile, as 1.15 to 1 very nearly; hence we have 3033.5 × 1·15=3488.5 English miles, for the direct distance between Liverpool and Halifax. Note.-The above process is not precisely that which is indicated by the formula; but it is tantamount to it, and saves the trouble of a subtraction in taking the complements of the latitudes, for it is obvious, that the sine of the complement is the same as the cosine, and the cosine of the complement the same as the sine. Longitude of Cape Churchill...... 93° 12' 0" West. East. Nat. num. .......................................................... 0.32927 sub. +0.64395 Nat. cos. b=+0.31468 EXAMPLE 2.-What is the direct distance between the Observatory of Paris, in latitude 48° 50′ 12′′ North, and longitude 2° 20′ 30′′ East, and Cape Churchill, in latitude 58° 48′ North, and longitude 93° 12′ West ? Here the latitudes are both north, and the longitudes one east and the other west; consequently, the sum is the angle contained between the meridians of the places; and being greater than a right angle, it agrees with the second case. Hence it is log. cos. 8.984840 log. 9.517555 22° 54' 42" South, and longitude 43° 9' West? Here the latitudes are the one North and the other South, and the longitudes both West; and since their difference is less than 90°, the example falls under the third case. Hence it is, log. cos. 9.883244 the latitude is 12° 33' North, and longitude 53° 23' East? In this example both the latitudes and longitudes are of different names, and the sum of the longitudes is greater than 90°; the example therefore falls under the fourth case, and is thus resolved : log. cos. 9.056071 22 54 42 log. cos. 9.964310 log. sin. 9.590297 ... Nat. num.............................................................. -0.10230........ log. 9.009878......... log. 8.927840 nat. uum. -0·08469 -0.08469 add. Nat. cos. b -0.18699; therefore, we have, b—180°—79° 13′ 20′′-100° 46′ 40′′-6046-66 geographical miles, or 6953 English miles, the distance sought. Since these four examples illustrate all the cases of which the problem is susceptible, it is only necessary that the reader should pay particular attention to the signs of the quantities as they arise in these cases, to avoid any error that may result from the different relations of the angular magnitudes. rack-pulley, n, in the same way as in ordinary roller-blinds; but when it is 7/2 a BAILLIE'S PATENT TRANSPARENT SLIDEVALVE VENTILATOR. In the ordinary glass louvre ventilators each of the louvres turns on a separate axis, and as long as the whole keep in equally perfect working order, the whole may be opened to any uniform extent, or altogether closed. It rarely happens, however, that the whole remain for any length of time in the same trim; some will not turn as freely, or to the same extent, as others; while one or two will not move at all, either one way or another; the ventilation becomes, in consequence, proportionally inefficient; the streams of air enter at different, often very conflicting angles; and permanent draughts are established where occasional currents only were wanted. In the present ventilator we have all the advantages of the glass louvres, free from any of these objections. It consists, firstly, of a series of louvres, which are permanently fixed at a certain inclination, so that the currents of air may be deflected upwards in one uniform direction ; and secondly, of a sliding valve, likewise of glass, by which the quantity of air admitted may be regulated at pleasure, and which, when closed, renders the openings perfectly air-tight. The whole is contained in a neat frame, which may be readily adapted by a common glazier to any of the panes of a window. αα Fig. A is an elevation and a vertical section of a sash-window, with a ventilator of this description, fixed in the position which experience has proved to be the best for avoiding draughts. are the fixed glass louvres; bc, the slide valve for regulating the quantity of air admitted, which is moved by the cords de, coinciding with, and hidden by, the sash-bars, and passing over pulleys (as at fg hj) to any required position. The cord is represented in the figure as being finally passed over a THE BRITISH ASSOCIATION-CAMBRIDGE MELTING, 1845. indicated arise from having the louvres stationary, instead of being moveable. For example:-First, the draught of cold air is avoided, which, in the case of moveable louvres, enters through the intervals that are required to be left between their ends and the sides of the frame. Secondly, the apparatus has no joints, nor other working parts, where the dust can accumulate and become hardened, so as to obstruct their action. It may be closed in a perfectly air-tight manner, even in the most dusty situations. Thirdly, its construction is so simple, that nothing but rough usage can injure it; and if out of order, it may be repaired by any ordinary workman. And fourthly, the cord or line by which the sliding valve is opened and shut (when such is used) may be carried to any part of a room, such as the bed-side, in the same manner as a bell-rope. We have had one of these ventilators fixed for some time in our own office, and should be wanting in thankfulness for the comfort it has afforded us, were we not to recommend it warmly to our friends. BRITISH ASSOCIATION-CAMBRIDGE Sir John Herschel, President.. The Antarctic Voyage of Sir James Ross has conferred a most important accession to our knowledge in the striking discovery of a permanently low barometric pressure in high south latitudes over the whole Antarctic ocean a pressure actually inferior by considerably more than an inch of mercury to what is found between the tropics. A fact so novel and remarkable will of course give rise to a variety of speculations as to its cause; and I anticipate that it will furnish one of the most interesting discussions which have ever taken place in our Physical Section. The voyage now happily commenced under the most favourable auspices for the further prosecution of our Arctic discoveries under Sir John Franklin, will bring to the test of direct experiment a mode of accounting for this extraordinary phenomenon thrown out by Colonel Sabine, which, if realized, will necessitate a complete revision of our whole system of barometric observation in high 9 latitudes, and a total reconstruction of ail our knowledge of the laws of pressure in regions where excessive cold prevails. This, with the magnetic survey of the Arctic seas, and the not improbable solution of the great geographical problem which forms the chief object of the expedition, will furnish a sufficient answer to those, if any there be, who regard such voyages as useless. Let us hope and pray that it may please Providence to shield him and his brave companions from the many dangers of their enterprise, and restore them in health and honour to their country.-The President's Address. Repeal of the Glass Duty. A very great obstacle to the improvement of telescopes in this country has been happily removed within the past year by the repeal of the duty on glass. Hitherto, owing to the enormous expense of experi ments to private individuals not manufacturers-and to the heavy excise duties imposed on the manufacture, which has operated to repress all attempts on the part of practical men to produce glass adapted to the construction of large achromatics, our opticians have been compelled to resort abroad for their materials-purchasing them at enormous prices, and never being able to procure the largest sizes. The skill, enterprise, and capital of the British manufacturer have now free scope, and it is our own fault if we do not speedily rival, and perhaps outdo, the far-famed works of Munich and Paris. Indeed, it is hardly possible to overestimate the effect of this fiscal change on a variety of other sciences to which the costliness of glass apparatus has been hitherto an exceeding drawback, not only from the actual expense of apparatus already in common use, but as repressing the invention and construction of new applications of this useful material.-The President's Address. Magnetic Machines. Dr. SCORESBY described a large magnetic machine which he had constructed, with some results of its action. The principal part of the machine consists of two cases, or fasciculæ of magnetic bars, of unusually large dimensions, on principles which may be thus summarily stated: 1. That magnetic bars designed for large combinations may be conveniently constructed of various pieces; that the separation of a long bar, say of three or four, into several portions, is not disadvantageous in regard to power, and that the resulting power is similar, whether in the combining of several series of short bars the elementary bars be of the same or of unequal lengths. 2. That the relative powers of magnets, whether single or com |