From these circumstances it appears, that water, by being placed in a new situation free from receiving heat from other bodies, and exposed in large surfaces to the air, may be brought to freeze when the temperature of the atmosphere is some degrees above the freezing point on the scale of Fahrenheit's thermometer; and by being collected and amassed in a large body, is thus preserved, and rendered fit for freezing other fluids, during the severe heats of the summer season. In effecting which, there is also an established mode of proceeding; the sherbets, creams, or whatever other fluids are intended to be frozen, are confined in thin silver cups of a conical form, containing about a pint, with their covers well luted on with paste, and placed in a large vessel, filled with ice, saltpetre, and common salt, of the two latter an equal quantity, and a little water to dissolve the ice and combine the whole. This composition presently freezes the contents of the cups to the same consistency of our ice creams, &c. in Europe; but plain water will become so hard as to require a mallet and knife to break it. On applying the bulb of a thermometer to one of these pieces of ice, thus frozen, the quicksilver has been known to sink 2 or 3 degrees below the freezing point; so that from an atmosphere apparently not cold enough to produce natural ice, ice shall be formed, collected, and a cold accumulated, that shall cause the quicksilver to fall even below the freezing point. XXIII. Of the House-Swallow, Swift, and Sand-Martin. By the Rev. Gilbert White. p. 258. This excellent paper is reprinted in Mr. White's History of Selborne, to which the reader is referred. XXIV. Of a Machine for raising Water, executed at Oulton, in Cheshire, in 1772. By Mr. John Whitehurst. p. 277. Presuming the mode of raising water by its momentum may be new and useful to many individuals, Mr. W. was induced to send a description of a work, executed in 1772, at Oulton, Cheshire, the seat of Philip Egerton, Esq. for the service of a brewhouse and other offices, which was found to answer effectually. The circumstances attending this water-work require a particular attention, and are as follow. A, in fig. 10, pl. 12, represents the spring or original reservoir, its upper surface coinciding with the horizontal line BC, and with the bottom of the reservoir K. D the main pipe, 1 inch diameter, and nearly 200 yards in length. E a branch pipe, of the same diameter, for the service of the kitchen. offices, situated at least 18 or 20 feet below the surface of the reservoir a; and the cock F was about 16 feet below it. G represents a valve-box, g the valve, H an air vessel, oo the ends of the main pipe inserted into н, and bending downwards, to prevent the air from being driven out when the water is forced into it w the surface of the water. Now it is well known, that water discharged from an aperture, under a pressure of 16 feet perpendicular height, moves at the rate of 32 feet in a second of time; therefore such will be the velocity of the water from the cock F. And though the aperture of the cock F is not equal to the diameter of the pipe D, yet the velocity of the water contained in it will be very considerable: consequently, when a column of water, 200 yards in length, is thus put into motion, and suddenly stopped by the cock F, its momentous force will open the valve g, and condense the air in н, as often as water is drawn from F. In what degree the air is thus condensed, is needless to say in the instance before us; therefore Mr. W. only observes, that it was sufficiently condensed to force out the water into the reservoir K, and even to burst the vessel H, in a few months after it was first constructed, though apparently very firm, being made of sheet lead, about 9 or 10 pounds weight to a square foot. Whence it seems reasonable to infer, that the momentous force is much superior to the simple pressure of the column IK; and therefore equal to a greater resistance, if required, than a pressure of 4 or 5 feet perpendicular height. It seems necessary further to observe, that the consumption of water in the kitchen offices is very considerable; that is, that water is frequently drawing from morning till night all the days of the year. XXV. On Occultations of Stars and Geometrical Theorems. Being an Extract of a Letter from Mr. Lexel, to Dr. Morton. Dated Petersburg, June 14, 1774. p. 280. As I propose, says Mr. L., to make some researches concerning the difference of the meridians of the principal Observatories of Europe, which I am persuaded can best be ascertained by the occultations of the fixed stars by the moon ; it would be of great service to me to be furnished with the observations that have been made, or that will be made, this year, of the occultations of a or of γ Tauri by the moon. I beg therefore Sir, you will please to desire Mr. Maskelyne to communicate them to me, towards the beginning of the next year, directed to Mr. Euler, secretary of our Academy. It would also be of great use to me to have the observation of the occultation of the Pleiades by the moon the 15th of March, 1766, in case it has been taken at Greenwich. The following are some observations of Mr. Wargentin, of the occultations of andy Tauri. 1773, Nov. 1.... 11h 56m 12'.... Emersion of a, uncertain to some seconds. .... 7 15 51 .... Emersion, Feb. 18.... 6 39 51 .... Immersion of y, very certain. Emersion, within 2 seconds. 16....10 21 31. May 22.... 13 2 ..... Emersion of a almost certain; the immersion was not {Emererved on account of clouds. Immersion, Emersion, } both certain. Immersion of «, very certain. .Emersion of the same. .. Immersion of Flamstead's 115 in 8. .... Immersion of a star of the 6th magnitude in II. 20........ Immersion of m Virginis, very certain. I have lately discovered two curious. theorems, which I shall here communicate to the R. S. Theorem.-Let A, B, C, D, E, F, be a polygon whose sides are named a, b, c, d, e, f; and the exterior angles a, ß, y, d, e, C, so that the side a be placed between the angles a and ß, b between ß, y, &c. 1. a × sin. a + b × sin. (a + ß) + c × sin. (a + ß + y) + dx sin. (a+ß+y+d+ex (sin.a + ß + y + d + :) +fx sin. (a +ß+y+d+ε + 5) = 0. B A D E 2. a x cos. a+b× cos. (a +ß) + cx cos. (a+ß+ y) + dx cos. (a +ß+y+♪ +ex cos. (a +ß+y+d+ 1) +ƒ× cos. (a + ß+y+d+e+() = 0. = In fact it is sin. (x + ß + y +♪ + + () = sin. 360° 0, and cos. (a +ß +y+d+e+() = + 1; but in order to give the same form to the two expressions, I rather chose to represent them as I have done. By means of these two theorems the solution of polygons will be as easy as that of triangles by common trigonometry. XXVI. Investigation of a General Theorem for finding the Length of any Arc of any Conic Hyperbola, by Means of two Elliptic Arcs. With some other New and Useful Theorems deduced from it. By J. Landen, F.R.S. p. 283. 1. From the theorem noticed in art. 1 of the author's paper in the Philos. Trans., 1771, (p. 150, of this abridged vol.) it follows, that in the hyperbola AD (pl. 12, fig. 11), if the semi-transverse axis AC bem-n; the semi-conjugate =2√mn; and the perpendiclar CP, from the centre c on the tangent DP, √ (m — n)2 — t2); the difference DP- AD, between the said tangent DP and 2. It is well known, that in any ellipsis whose semi-transverse axis is m, and semi-conjugate n; if x be the abscissa, measured from the centre on the transverse axis, and z the arc between the conjugate axis and the ordinate corresponding to x,m2. pears, that in the ellipsis aed (fig. 12) whose semi-transverse axis cd is m+n, semi-conjugate ca = 2√mn, and abscissa cb (corresponding to the ordinate be) m + n = m-n (m + n)2 t2 (mn) t Xi. t; the arc ae is equal to the fluent of ✔ 3. In the ellipsis aefd (fig. 13), the semi-transverse axis cd being = m; the semi-conjugate ca = n; and the abscissa cb (corresponding to the ordinate be) = x; if ep, the tangent at e, intercepted by a perpendicular (cp) drawn to it from the centre c, be denoted by t; gx X V. (as is well known) will be =t, g being as in the preceding article. m2 m2 r2 gr m2g + t2 ‚ (m2 — n2)2 — 2 × (m2 + n2) × t2 + t* ̧ 2g (m2 + n2) × it — t3¿ √[2g (m2 —n2)2 — 2 × (m2 + n2) × t2 + t‘} But being = √ that m2 × x is = %. (m2 + n^)2 xi-ti 2g it + 2g From which equa (m2 + n2) × i - ri √ [og(m—n) 3—t3 × (m+n)2 — t2)] ' - Xi, as observed in the preceding article, it appears It is obvious therefore that ¿ is = i + ¦ × ✅[((m — n)1 — t1) × ((m + n)2 — t2)] (m + n)2 × i — t'i - (m + n)2 — t3 (m + n)1t2 t2 Xi. Whence, taking the fluents by the theorems in art. 1 and '(fig. 11)+=(fig. 12); consequently the hyperbolic arc AD is DP +ae + 2t4ae. Thus, beyond my expectation, I find, that the hyperbola may in general be rectified by means of two ellipses. Writing E and F for the quadrantal arcs ad, ad (fig. 12 and 13) respectively, and L for the limit of the difference DP AD, while the point of contact (D) is supposed to be carried to an infinite distance from the vertex A of the hyperbola (fig. 11), we find 2F — EL, the value of ae being = +r++m➡ in when t is = mn; that is, when e coincides with d (fig. 12), and P with c (fig. 11), by what I have proved in the before-mentioned paper, art. 10. 4. From what is done above, the following useful theorems are deduced. These theorems still refer to fig. 11, 12, 13; but now the values of the several lines in them (being not as before) are as here specified, viz. Fig. 11, in the hyperbola AD, the semi-transverse axis ac is now = a; the semi-conjugate = b; the perpendicular CP, from the centre c on the tangent DP, is = √ az, the said tangent DP = √ ÷ × (b2 + 2kz √2 × (b2 + 2kz — z2); and the abscissa CB (corresponding to the a Fig. 2. In the ellipses aed, the semi-transverse axis cd is = √(a2 + b2); the semi-conjugate ca = b; the abscissa cb = √ a2 + b2 × √ (a — z) ; and the ordinate be=¿ √%. a Fig. 13. In the ellipsis aefd, the semi-transverse axis cd is = √(a2 × b2) + a; the semi-conjugate ca = √ (a2 + b2) a; the tangents ep, fq, intercepted by perpendiculars (cp, cq) drawn to them from the centre c, each = √ a x (az); and the abscissa (cb' or cb") on cd, corresponding to the point e or f, √[√(a2+b2)+a−2+√(23+−2)] √2+ √(a2—b2) of the curve is determined by the expression X cd. The quadrantal arc ad (fig. 12) is denoted by E; and the quadrantal arc ad (fig. 13) is denoted by F; L the limit of DP — AD (fig. 11) is = 2F — E. From what is now done, I might proceed to deduce many other new theorems, for the computation of fluents; but I shall at present decline that business: and, after giving a remarkable example of the use of theorem 4, in computing the descent of a heavy body in a circular arc, conclude this paper with a few observations relative to the contents of the preceding articles. 5. Let lpqn (fig. 14) be a semi-circle perpendicular to the horizon, whose highest point is 1, lowest n, and centre m. Let ps, qt, parallel to the horizon, meet the diameter Imn in s and t: and let the radius Im (or mn) be denoted by r, the height ns by d; and the distance st by x. Then, putting h for (16 feet) the space a heavy body, descending freely from rest, falls through in one second of time; and supposing a pendulum, or other heavy body, descending by its gravity from p, along the arc pqn, to have arrived at q; the fluxion of the time of descent will be = √[2rd — d2 - 2 (r — d) (d — x)]° The fluent of which, or the time of descent from p to q is, by theorem 4 of the preceding article, = x de ef; a (in that theorem) being taken =√ d, b =√(2r — a), 2r hx (2r-d) 2r cb (fig. 2) = √2 × √ (d — x), and ep, fq, (fig. 13) each =√ (d — x). Hence 2r it appears, that the whole time of descent from p to n is = √hx (2r-d) X (E F); when, in fig. 12 and 13, the semi-axes are taken according to the values of a and b just now specified. |