wore all the winter a black patch over it, to guard against fresh cold. The cyst, when first taken away, measured 34 inches in length, 14 in diameter, and contained a large cup-full of water. It appeared to be the tunica sclerotica, was of a clear pellucid white, and of so delicate a texture, as scarcely to admit of being touched without tearing; and when dried with all possible care, became so brittle, that Mr. Hopkins could hardly preserve it in the manner he had sent it. Remarks.-In both S. Earle's and J. Law's cases, the eye was distended by the accumulation of the aqueous humour, separated in great quantity by the repeated straining of the blood-vessels in the hooping cough, which might gradually relax and enlarge the aqueous ducts of S. Earle's eye; and possibly by the rupture of those ducts, and of some blood-vessels, at the time J. Law exerted himself violently in beating dung about the close; for in either case the impetus of the blood must have been so violent as to produce those effects. However, from the hydrophthalmia succeeding the operation on Law, the fungous excrescence, and continual serous discharge during several months from the wound, it plainly appears that an abundance of aqueous humour was discharged at first by the distension or laceration of the aqueous ducts, and latterly for want of a contraction of those vessels, and of the lymphatics, which were no longer of use. Both these cases showed the necessity of inquiring particularly into the causes of diseases of the eyes, as well as of other parts of the body; for by barely attending to the symptoms, the disease will not be removed, though the symptoms be alleviated. Bleeding, and moderate evacuations, would doubtless at first have. decreased the tension and pain, and assuaged the inflammation; but both topical applications, and internal medicines, were properly to be adapted, and a suitable diet regulated. Not to mention the absurd and impertinent abuse of empirics, what benefit could accrue, in both these cases, from unctuous, laxative, or emollient applications, from drastic and mercurial purges? Though such appli cations might be well intended, to take off the tension and inflammation; yet as the distension of the blood-vessels only increased gradually, as the globe of the eye was enlarged; so whatever application relaxed the coats of the eye must infallibly stretch out the vessels yet farther, and cause a greater pain and inflammation; which drastic and mercurial purges would also increase. The only method then to be pursued in such bad cases would be at first to endeavour to remove the fulness of the blood, and make use of such topical remedies as would contract without irritation. If the cause remains, as the hooping cough in S. Earls case, no amendment of the eye can be expected, while the patient's bloodvessels are continually strained by frequent coughing. This illness, therefore should be attended to, and removed as soon as possible. But should the eye be so enlarged as to protrude itself out of the orbit, there seems no other way to lessen the bulk of the eye, than by making a puncture. with a proper instrument, to let out the aqueous humour; and then apply such agglutinant and contracting collyria, as may reduce the distended coats and vessels to their former size. This operation should be performed before the humours are vitiated, the sight lost, the vessels in a state of suppuration, and the coats of the eye too far extended; for at that time nothing less than extirpation can be of use. Professor Nuck, in his Tractatus de Ductibus Oculorum Aquosis, p. 120, relates the success he had in curing a young man by 5 repeated punctures, and a strict observance in a proper use of all the non-naturals. CII. On the Heat of the Weather in Georgia. By H. Ellis, Esq. Governor of Georgia, and F. R. S. p. 754. One cannot here sit down to any thing that requires much application but with extreme reluctance; for such is the debilitating quality of our violent heats at this season (July), that an inexpressible languor enervates every faculty, and renders even the thought of exercising them painful. It is now (writes Mr. Ellis) about 3 o'clock; the sun bears nearly s.w., and I am writing in a piazza, open at each end, on the N.E. side of my house, perfectly in the shade: a small breeze at s.E. blows freely through it; no buildings are nearer to reflect the heat than 60 yards: yet in a thermometer hanging by me, made by Mr. Bird, and compared by the late Mr. George Graham with an approved one of his own, the mercury stands at 102. Twice it has risen this summer to the same height; viz. on the 28th of June, and the 11th of July. Several times it has been at 100, and for many days successively at 98; and did not in the nights sink below 89. It is highly probable that the inhabitants of this town breathe a hotter air than any other people on the face of the earth. The greatest heat we had last year was but 92, and that but once: from 84 to 90 were the usual variations; but this is reckoned an extraordinary hot summer. The weather-wise of this country say it forebodes a hurricane; for it has always been remarked, that these tempests have been preceded by continual and uncommon heats. I must acquaint you however that the heats we are subject to here are more intense than any other parts of the province, the town of Savannah being situated on a sandy eminence, and sheltered all round with high woods. Yet it is remarkable. that this very spot, from its height and dryness, is reckoned equally healthy with any other in the province. in I have frequently walked 100 yards under an umbrella, with a thermometer suspended from it by a thread to the height of my nostrils, when the mercury has risen to 105; which is prodigious. At the same time I have confined this instrument close to the hottest part of my body, and have been astonished to observe that it has subsided several degrees. Indeed I never could raise the mercury above 97 with the heat of my body. I have traversed a great part of this globe, not without giving some attention to the peculiarities of each climate; and I can fairly pronounce that I never felt such heats any where as in Georgia. I know experiments on this subject are extremely liable to error; but I presume I cannot now be mistaken, either in the goodness of the instrument, or in the fairness of the trials, which I have repeatedly made with it. This same thermometer I have had thrice in the equatorial parts of Africa; as often at Jamaica, and the West India islands; and on examination of my journals, I do not find that the quicksilver ever rose in those parts above the 87th degree, and to that but seldom its general station was between the 79th and 86th degree; and yet I think I have felt those degrees, with a moist air, more disagreeable than what I now feel. hu u my account of the late expedition to the north-west, I have observed that all the changes and variety of weather that happen in the temperate zone throughout the year, may be experienced at the Hudson's Bay settlements in 24 hours. But I may now extend this observation; for in my cellar the thermometer stands at 81, in the next story at 102, and in the upper one at 105; and yet these heats, violent as they are, would be tolerable but for the sudden changes that succeed them. On the 10th of December last the mercury was at 86; on the 11th it was so low as 38 of the same instrument. What havock must this make with a European constitution? Yet but few people die here out of the ordinary course; though indeed one can scarcely call it living, merely to breathe, and trail about a vigorless body; yet such is generally our condition from the middle of June to the middle of September. CIII. The Invention of a 2d, 3d, 4th, or 5th, &c. whole Series being known. General Method for determining the Sum of every As the doctrine of series is of very great use in the higher branches of the mathematics, and their application to nature, every attempt tending to extend that doctrine may justly merit some degree of regard. The subject of the present paper will be found an improvement of some consequence in that part of science. And how far the business of finding fluents may, in some cases, be facilitated by it, will appear from the examples subjoined, in illustration of the general method here delivered. The series propounded, whose sum (s) is supposed to be given (either in algebraic terms, or by the measures of angles and ratios,. &c.) is here represented by a + bx + cx2 + dx3 + ex1 &c. and Mr. S. first gives the solution of that case, where every 3d term is required to be taken, or where the series to be summed is a + dx3 + gx+hx &c. By means of which, the general method of progxa kxo ceeding, and the solution of other case, will appear evident. every Here then every 3d term being required to be taken, let the series (a + dø3 +ga &c.) whose value is sought, be conceived to be composed of 3 others. + × (a + b × px + c × p2x2 + d x p3x3 + ex p1x1 &c.) ÷ × (a + b × qx + c × g2x2 + dx q3 3 + e × 9*x* &c.) 2 × (a + bx rx + ex r2x2 + d x 3 3 + ex r11 &c.) having all the same form, and the same coefficients with the series first proposed, and where the converging quantities px, qx, ra, are also in a determinate (though yet unknown) ratio to the original converging quantity x. Now in order to determine the quantities of these ratios, or the values of p, q, and r, let the terms containing the same powers of x, in the two equal values, be equated in the common way. So shall, 4.4 And consequently, ÷ ex p2x2 + ÷ e × q2x2 + je × r1x = 0, &c. Now make p3 1, q3 = 1, and 73= 1; that is, let p, 6 = 5 p + q + r = 0 p1 + q* + r1 = 0, &c. q, and r, be the three roots of the cubic equation 23 = 1, or z3 — 1 = 0: then, seeing both the 2d and 3d terms of this equation are wanting, not only the sum of all the roots (p+q+r) but the sum of all their squares (p2 + q2 + r2) will vanish, or be equal to nothing, by common algebra, as they ought to fulfil the conditions of the first two equations. Also, since p3 = 1, q3 = 1, and 73 = 1, it is also evident, that p1 + q1 + p2 (= p + q + r) = 0, p3 + q3 + p3 (= p2 + q2 + p2) =0, po + qo + 7.o (= p3 + q3 + r·3) = 3. Which equations being, in effect, nothing more than the first 3 repeated, the values of p, q, r, above assigned, equally fulfil the conditions of these also: so that the series arising from the addition of 3 assumed ones will agree, in every term, with that whose sum is required: but those series, of which the quantity in question is composed, having all of them the same form and the same coefficients with the original series a + bx + cx2 + dx3 &c. (= s), their sums will therefore be truly obtained, by substituting pr, qx, and rx, successively, for x, in the given value of s. And, by the very same reasoning, and the process above laid down, it is evident, that, if every nth term, instead of every 3d term, of the given series be taken, the values of p, q, r, s, &c. will then be the roots of the equation 2" - ] = 0;* and that the sum of all the terms so taken, will be truly obtained by substituting 360° 360* 360° 2 x 3 X n n n * If a, ß, v, d, &c. be supposed to represent the co-sines of the angles &c, the radius being unity; then the roots of the equation z 10 (expressing the several values of p, q, r, s, &c.) will be truly defined by 1, a + √au — 1, a — √ au ~ 1, B + √ßß — 1; ß — BB-1, &c. The demonstration of this will be given farther on-Orig. px, qx, τx, sx, &c. successively for x, in the given value of s, and then dividing the sum of all the quantities thence arising by the given number n. The same method of solution holds equally, when, in taking every nth term of the series, the operation begins at some term after the first. For all the terms preceding that may be transposed, and the whole equation divided by the power of x in the first of the remaining terms; and then the sum of every nth term, beginning at the first, will be found by the preceding directions; which sum, multiplied by the power of x that before divided, will evidently give the true value required to be determined. Thus, for example, let it be required to find the sum of every 3d term of the given series a + bx + cx2 + dx‹3 + ex1, &c. (=s), beginning with cx2. Then, by transposing the first two terms, and dividing the whole by x2, we shall have c + dx + ex2 + ƒx3 &c. s-a bx xx (s). From which having found the sum of every 3d term of the series c + dx + ex2 + ƒx3 &c. beginning at the first c, that sum, multiplied by x2, will manifestly give the true value sought in the present case. And here it may be worth while to observe, that all the terms preceding that at which the operation, in any case, begins, may (provided they exceed not in number the given interval n) be entirely disregarded, as having no effect at all in the result. For if in that part (-) of the value of s', above exhibited, in which the first terms, a and b, enter, there be substituted pr, qx, rx, successively, for x, according to the prescript, the sum of the quantities thence arising will be a a a xx b b q3= 1, &c. or p2 = - 2/2 × (p + q + r) = 4, a b x which, because p3 = 1, &c. may be expressed thus: − × (p2 + q2 + r2). But, that p+q+r=0, and p2 + q2 + r2 = 0, has been already shown; whence the truth of the general observation is manifest. Hence it also appears, that the method of solution above delivered, is not only general, but includes this singular beauty and advantage, that in all series whatever, the terms of which are to be taken according to the same assigned order, the quantities (p, q, r, &c.), by which the solution is performed, will remain invariably the same. The greater part of these quantities are indeed imaginary ones; and so likewise will the quantities be that result from them, when substitution is made in the given expression for the value of s. But by adding together, as is usual in like cases, every two corresponding values, 30 resulting, all marks of impossibility will disappear. If, in the series to be summed, the alternate terms, viz. the 2d, 4th, 6th, &c. should be required to be taken under signs contrary to what they have in the original series given; the reasoning and result will be nowise different; only instead of making p3 + q3 + r3, or p” + q′′ + 1", &c. = + 3 orn, the |
