one inch deep of rain is equal to 28-274 cubic inches (nearly one pint wine measure), and in weight at the temperature of 60° to 7139.1850 grains ; consequently { inch = 1784.7962 or 1784 grains nearly. Let, therefore, the glass jar be exactly balanced in good scales, and then filled with rain or distilled water to the perfect equipoise of 1784 grains, marking the height on the side of the measure, which will be the indication of 25 or inch of rain in the guage. Were the cylinder of equal diameter throughout, it would only remain to divide the space so marked off into 25 equal parts ; but as these glasses are seldom quite uniform, it is necessary to check the measure by weight to every .05 of an inch, which is easily done by weighing with 1427 grs. for .20, with 10704 for 15, with 714 for.10, and with 357 grs. for •05 of an inch, marking these divisions severally and accurately on the side of the measure, and then dividing each into five equal parts, observing to allow in the lowest for the bulb usually found at the bottom. Thus the measure will be graduated to the hundredths of an inch, and if it be about 1; inch in diameter, the spaces will be large enough to halve, so that the register may be conveniently kept in three places of decimals; thus instead of .10', set down •105, being so many thousandths of an inch. If there is any difficulty to cut the glass, the graduation may be marked with a pen on a slip of paper pasted on the side of the measure, observing that it should be quite dry before the operation takes place. It is thus easy to provide an accurate pluviameter; but to find a suitable situation for fixing it " at a sufficient distance from trees, buildings, or any object that might obstruct” the free current of the wind, is a matter of great difficulty, and of the greatest importance. For this reason, the tops of the highest buildings have been heretofore selected. There is, however, cause to suspect that they are the most ineligible. In the reports from Kinfaun's Castle, the upper is made to indicate more rain than the lower guage. This is contrary to all similar observations, and to the nature of things. The error must, therefore, be in the different capacity or situation of the instruments used. To this phenomenon, which has excited so much speculation, our observations have been particularly directed. Three guages and measures, exactly alike in form and size, have been used. The first was already fixed on the top of onr Museum (from this our annual reports are drawn), higher than the level of the adjacent chimney stacks, and consequently free from lateral obstrůttions. The house is open in front, joined on each side to others, and its back towards the continuous buildings of the town. The guage is 45 feet above the surface of the ground, and 143 above the level of the sea. The two other guages are fixed at the level of the ground, each in a garden, and at a distance of about 60 yards from any building or high trees, being in respect of all circumstances, that seem to affect the fall of rain, similar each to the other. The first of these two, or No. 2, is 150 yards distant from the Museum, and 90 feet above the sea. No. 3 is about 500 yards from both the former, and 70 feet above the level of the sea. In these two, viz. No. 2 and No. 3, the quantity of rain was on the whole equal, only varying occasionally in a small degree above or below each other, but the difference between them and No. 1 is very great, viz, above 3 to 2, the result of 12 months being, in No.1, 30.475 inches ; and in No. 2 and No.3, 46.080 inches. The ratio varied considerably in several months, as for instance, the total of No. 2 and 3. Aug. 1821, was in. 4.470 September. 4.520 October. 5.770 November 5.170 December 9.500 No. 1, 9 6 : 3.550 11 7 3.110 10 6 6.480 19 13 These months were unusually wet, and the three last remarkably stormy: Having observed that the difference between the first and the other guages varied with the more or less wind, its velocity has been registered from observation ; but not having an accurate anemometer, we cannot yet offer any certain conclusion further than this, that the difference in the quantity of rain received in a guage placed on the top of a building and one at a level with the surface of the ground, is, for some reason or other, proportioned to the velocity of the wind; and that the average excess of the lower guage is much greater than can be attributed to any or all the causes hitherto assigned. For admitting all that can be due to the difference of the sine of the angle of inclination at which the falling drops may reach the earth ; and also all that could accrue from a continued condensation of aqueous vapour between the altitude of the upper guage and the surface of the ground, yet the aggregate of both would, in an elevation of only 45 feet, be trivial, in comparison of the enormous difference found every month, and on the average of the whole year. The facts obtained do not yet, perhaps, warrant the positive conclusion, and we, therefore, offer it only as a conjecture that the aforesaid difference is owing chiefly to the whirl or eddy occasioned by the recoil of the gusts of wind striking on the sides of the building—an effect very visible in the disturbance of smoke issuing from chimneys during a high wind. Since the discovery of the self-registering thermometer, and the consequent notation of the daily maximum and minimum of temperature, it has been found that the annual mean heat of the north and south of Great Britain is much more equal than was supposed ; and it seems probable that the annual mean of the . the Quantity of Rain falling in adjacent Places. 21 fall of rain will be found to differ much less than .hitherto recorded. Nothing will contribute more to ascertain the fact than an uniformity of guages, as well in situation, as size and shape. It is desirable that in all cases there should be a guage on the level of the surface of the ground. A pit must be prepared just fitted to the bottle in which it may so stand that the edge of the funnel shall be but half an inch above the surface, and care taken that the rim of the basin be truly horizontal. If any obstacle to the free course of the wind occurs 'within 100 or 200 yards of the guage, its height, breadth, and direction, should be noticed; and in respect of those placed on the tops of buildings, the length of pipe between the funnel and receiver, and whether within or without the house, should be mentioned; as well as the height of the guage above the ground, and above the level of the It is also desirable that the barometrical tables should be always reduced to the temperature of 32°, or, if not, that the omission should be stated; and the thermometrical tables which give the true maximum and minimum of every 24 hours are preferable to the observations of fixed periods, which very often fail to show either. I am, Sir, your most obedient servant, H. BOASE. sea. ARTICLE VI. On Finding the Sines of the Sum and Difference of Two Arcs. By Mr. James Adams. Stonehouse, near Plymouth, June 8, 1822. It having occurred to me that by making a small alteration in the methods given by Mr. Leslie in his Geometry, and by Mr. Woodhouse in his Trigonometry, the demonstrations of the two fundamental formulæ for compound arcs may be rendered still more simple than those usually given, I will thank you to insert the following in the Annals of Philosophy, when convenient. I am, Sir, your most obedient servant, JAMES ADAMS. To find the Sine of the Sum of two Arcs. Let the quadrilaterals A B D E be inscribed in a circle and semicircle, whose centres are C, and diameters A E, the diagonals of which being A E, BD, in fig. 9 (Pl. XIII), and A B, D Е, in fig. 10. Bisect the arcs E B, ED, BD, in the points.t, r., w; and draw the radii Ct, C r, and Cw, which will (3 and 30.3.e.) bisect at right angles the chords E B, ED, B D, in the points m, v, n; then will (31.3.e.) the triangle A BE be similar to Cm E, and the triangle ADE similar to CvE; therefore (4.6.e.) A B is the double of Cm, and A D the double of C v. We then have in fig. 10, by Prop. D, Simson's Euclid, A B x D E + A D BE = AE X BD.... (a) Or, 2 Cm x 2 Ev + 2 Cv x 2 Em = 2 x 2 B n (radius unity.) Or, C m + E 6 + C o x E m = B n = D n. Therefore, cos. Et sin. Er + cos. Er sin. Et = sin, Bw = sin. r t = sin. (Er+Et). To find the Sine of the Difference of two Arcs. We have in fig. 2, by Prop. D. ibid. A D x B E + A E x B D = A B x D E. Therefore, A B XDE - ADXBE = A E x B D. From whence, see equation (a), we have cos. Et sin. Er - cos. Ersin. Et = sin. B w = sin. rt=sin. (Er - Et). If from A and E, the extremities of the diameters A E, the perpendiculars A x, E s, be drawn to the chords B D, we shall, by a method equally as simple as the preceding, be able to find the cosines of the sum and difference of two arcs. For by Prop. C. 6, Simson's Euclid, we have the following equations; víz. A B'X AD= A x x A E, and Es X AE = EB X E D in both the figures, from whence we get AB X AD-EB ED= AE (Ax-Es) =2CnX AE, fig. 1 (6) AB X AD+EBXED=AE (Ax+Es) =2Cnx AE, fig. 2 .. (c) Or, 2 Cm x 2 Cv - 2 Et x 2 Ev = 2 * 2 C n (radius unity). Or, Cmx Co-Etx Ev= Cn. Therefore, cos. Et cos. Er sin. Et sin. E r = cos. B w = cos. r t = cos. (Er + E t), by equation (b). In like manner, we have cos. Et cos. Er + sin. Et sin. Er = cos. B w = cos. r t = cos. (Er + E t), by equation (c). The arcs E B and E D, as well as their halves E t and Er, are supposed to be of the same magnitude in both the figures. From the foregoing, we readily obtain the following equations; viz. sin. (A + A) = sin. 2 A = sin. A cos. A + sin. A cos. A = 2 sin. A cos. A . cos. (A + A) = cos. 2 A = cos? A sin.' A.... 1 + cos. A sin. A 1 - cos. A sin. A sin.(A - A) = sin. ; A = sin. A cos. Į A sin. • A cos. A . (e) cos. (A – ŽA)=cos. A=cos. A cos. A + sin. A.. (f) From equation (e) we have sin. ¢ A (1 + cos. A) = sin. A cos. A; therefore, cot. A= From equation we have cos. A (1-cos. A) = sin. A sin į A; therefore, tan. A = ; by adding and subtracting the two last equations, we obtain tan. ¡ A + cot. į A = 2 cosec. A; and cot. A tan. ^ A = 2 cot. A. From whence cosec. A = cot. 1 cot. ¢ A + tan. A A cot. A = tan. į A + cot. A; and cot. § A tan. ğ A A; or = sec. 2 A. We also have by the common property of sines and cosines cos. A + sin. į A = 1, and by equation (d) cos. A - sin.e A = cos. A; by adding and subtracting the two last equations, we have sin. ; A = and cos. ; A = sec. cot. A + tan. A tan. A cos. A 2 On the Use of Tincture and Brazil Wood in distinguishing seve ral Acids, and on a new Yellow Colour obtained from it. By M. P. A. de Bonsdorff.* It is well known that Brazil wood, when treated with an alkaline solution, yields a very fine violet colour. It is on account. of this property that the tincture of Brazil wood, or paper coloured by it, is used in chemistry as a very delicate test of the alkalies. Besides this property, it possesses another which may prove interesting to the chemist; it may be seen by the experiments which I had occasion to make on this substance, and which are the subject of this memoir, that Brazil wood paper may be employed not merely as a delicate test of the presence of acids in general, but as a certain means of detecting several acids, and distinguishing them from each other. With respect to the action of acids upon the red colour of Brazil wood paper, it is to be observed; first, that a concentrated * From the Annales de Chimie et d: Physique, + The chemists of France and England prefer reddened litmus or turmeric paper to detect an excess of alkali; but these reagents, and especially the latter, cannot be com pared as to sensibility with Brazil wood paper. |