== same quantity must here be made = 3 orn. Whence p" being — — 1, q", &c. the values of p, q, r, &c. will in this case be the roots of the equation z" + 1 = 0. It may now be proper to set down an example or two of the use and application of the general conclusions above derived. First then, supposing the series, whose sum is given, to be + + 4 2n+1 x+2 m &c. H. log. 1 — x (= s); let it be hence required to find the sum of the series (TM+ + m x" + n &c.) arising by taking every nth term, beginning Here the terms preceding being transposed, and the whole equation divided by x", we have with that whose exponent (m) is any integer less than n. which value, let px, qx, rx, &c. be successively substituted for x (according to prescript) neglecting entirely the terms +, as having no effect at all in the x log. (1 — rx), &c. Which multiplied by am (the quantity that before divided). × log. (1 — px) — — × log. (1 — 9x) — — × log. (1 qm n times the quantity required to be determined. α rm But now to get rid of the imaginary quantities q, r, &c. by means of their known values a + √ aa—1, a — No αα- 1, &c. it will be necessary to observe, that as the product of any two corresponding ones (a +✔aa—1) × (a−√ aa— 1) is equal to unity, we may therefore write (a ✔ -1)" (=) instead of its equal, and (a+√aa—1)" (=q′′) instead of its equal by which means the two terms, wherein these two quantities enter, will stand thus ; — (α — √ aa − 1)TM × Log. (1—9x) — (a+ √∞∞−1)TM × Log. (1—rx.) 1 : But if A be assumed to express the co-sine of an arch (a,) m times as great as that (360°) whose co-sine is here denoted by a; then will ▲ —√a^ — 1 are known to express the fluxions of the circular arcs whose co-sines are x and x, it is evident, if those arcs be supposed in any constant ratio of 1 to n, that Whence, by taking the fluents, nx log. (x + √xx-1) or log. (x + √xx—1),= log. (x+xx-1;) xx-1;) and consequently (*+ √xx—1)n = x + √xx−1 : whence also seeing -x-1 is the reciprocal of x+1, and xxx-1 of x+xx-1, it is also, evident that (√xx −1)» = x−√xx-1. Hence, not only the truth of the above assumpOo VOL. XI. αα (x+ αα (a-Vax-1), and A+ √AA-1= substituted above, we thence get —a× [log. (1—qx)+log.(1−rx)]+✔AA—1 × [log. (1—qx) — log. (1 — rx)]; of which the former part (which, exclusive of the factor A, is hereafter denoted by м) is manifestly equal to—A X log. (1—qx) × (1-rx) by the nature of logarithms = — A × log. [1−(9 + r) x + qrx2] ▲ × log. (1 — 2 ax + xx) by substituting the values of q and r: which is now entirely free from imaginary quantities. But in order to exterininate them out of the latter part also, put y= log. (1-qx)-log. (1-ra;) then will y= + - qx rx -qx 1-rr √1-xxxx fluxion of a circular arch (N) whose radius is 1, and sine = 9-rxx •q + rx x + xx expresses the √1-ax xx 1-2 ax + xx ; con sequently y will be = −2✔—1 × N: which, multiplied by √a^—1, or its equal 1×1-AA, gives 2 √1 — AAX N; and this value being added to that of the former part, found above, and the whole being divided by n, we — AM + 2√] — ^^ thence obtain-^^, XN, or X (-co-s. a × M + sin a x 2 N) for that part of the value sought depending on the two terms affected with q and r. Whence the sum of any other two corresponding terms will be had, by barely substituting one letter or value for another: so that, will truly express the sum of the series proposed to be determined; ì, m', м", &c. being the hyperbolical logarithms of 1-2 ax X ax, 1—2ßx + xx, 1-2 yx +xx, &c. N, N', N", &c. the arcs whose sines are tion, but what has been advanced in respect to the roots of the equation z′′ — 1 = 0, will appear manifest. For if r±√xx-1 be put = %, then will " = x±√xx−1)" = x ± √xx — where assuming x = 1 co-s. 0 = co-s. 360° co-s. 2 × 360° = co-s. 3 x 360°, &c. the equation will become z2 = 1, or zn 1 = 0; and the different values of r, in the expression 0 360° 2x 360° (x±√xx 1) for the root z, will consequently be the co-sines of the arcs these arcs being the corresponding submultiples of those above, answering to the co-sine x (= 1.) In the same manner, if x be taken =-1= co-s. 180° co-s. 3 x 180° = co-s. 5 × 180°, &c. then will z = −1, or z′′ + 1 = 0; and the values of r will, in this case, be the co-sines of 180° 180° xm m here it may not be amiss to take notice, that the series + since the series m + m + n m + 2n n 1 From which, and the reasoning above may be very readily deduced. For, xm-1x m + 3n &c.) for this last fluent, is that which arises by changing the signs of the alternate terms of the former; the quantities p, q, 7, &c. will here, agreeably to a preceding observation, be the roots of the equation z" + 1 = 0; and consequently, a, ß, y, d, &c. the co-sines of the arcs 5 X &c. as appears by the foregoing note. So that, making a, a', a", &c. equal here to the measures of the angles X m, 3 x 180° n 180° n X m, 5 X xm, &c. the fluent sought will be expressed in the very same manner as in the preceding case; except that the first term, — log. (1 — x,) arising from the rational root p= 1, will here have no place. —x,) After the same manner, with a small increase of trouble, the fluent of may be derived, m and n being any integers whatever. Mr. S. now 1 ± 2x2 + x2n sets down one example, where the impossible quantities become exponents of the powers in the terms where they are concerned. The series here given is 1−x++ &c. = the number whose hyp. log. is — x, and it is required to find the sum of every n term beginning at the first. Here the quantity sought will, according to the general rule, be truly defined by the nth part of the sum of all the numbers whose respective logarithms are- px, - qx,-rx, &c. which numbers, if N be taken to denote the number whose hyp. log. =1, will be truly expressed by N-*, N−9%, N-*, &c. whence, by writing for p, q, r, &c. their equals 1, a +√∞∞ 1, α-1ax1, ß + √BB−1, B — √ BB B VB6-1, &c. and putting ά= √1—∞∞, B' = 1—ßß, &c. we shall have × (N~** +N-7* + N-* &c.) = into N-*+ n n N~~~ × (N−ά×√ ̄ ̄ + N2x√ —1) + N-Rx × (N− B'× √ πs + N3* √ + &c.) Bat N-∞ √=1+ Nix is known to express the double of the co-sine of the arch whose measure, to the radius 1, is áx. Therefore we have into N-+N× 2 co-s. άx+N-E x 2 co-s. ẞ'x, &c. for the true sum, or be determined. n value proposed to The solution of this case in a manner little different, I have given says Mr. S. some time since in another place; where the principles of the general method, here extended and illustrated are pointed out. I shall put an end to this paper with observing, that if, in the series given, the even powers of x, or any other terms whatever be wanting, their places must be supplied with cyphers; which, in the order of numbering off, must be reckoned as real terms. Observation of a Lunar Eclipse, made at Lisbon, July 30, 1757. By J. Chevalier, F. R. S. From the Latin. p. 769. At 9h 15m 18s Beginning of the penumbra. 9 22 24 Beginning of the eclipse, doubtful. CV. Singular Observations on the Manchenille Apple.* By J. A. Peyssonel, M.D., F.R.S. From the French. p. 772. The cruel effects of the manchenille are well known its milk, which the savages make use of to poison their arrows, makes the wounds mortal. The rain which washes the leaves and branches causes blisters to rise like boiling oil; even the shade of the tree makes those who repose under it to swell; and its fruit is esteemed a deadly poison. Of the following facts Dr. P. can vouch for the truth: One Vincent Banchi, of Turin in Piedmont, a strong robust man, and a soldier, of about 45 years of age, belonging to the horse, was a slave with the Turks 11 years, having been taken prisoner at the siege of Belgrade. He was overseer of Dr. P.'s habitation towards the month of July of the year 1756. He was one day walking on the sea-side, and seeing a great number of apples on the ground, was charmed with their beautiful colours and sweet smell, resembling that of the apple called d'apis : he ate of them ignorantly, and found they had a subacid taste; and having eaten 2 dozen, he filled his pockets, and came home eating the rest, till the negroes told him it was mortal. About an hour afterwards his belly swelled considerably, and he felt a consuming fire in his bowels. He could not keep himself upright; and at night the swelling of his belly increased, with the burning sensation of his bowels. His lips were ulcerated with the milk of the fruit, and he was seized with cold sweats: but the principal negro made him a decoction of the leaves of a ricinus in water, and made him drink plentifully of it, which brought on a vomiting, The manchenille or mancaneel-tree is the hippomane mancinella of Linneus: it is elegantly figured in the Appendix to Catesby's Carolina. + Avellana purgatrix; in French, medicinier.-Orig. followed by a violent purging; both which continued for 4 hours, during which it was thought he would die. At length these symptoms became less; and the negroes made him walk, and stir about by degrees; and soon after they were stopped. Rice-gruel, which they gave him, put an end to all these disorders; and in 24 hours he had no more ailments nor pain; the swelling of his belly diminished in proportion to his evacuations upward and downward, and he had continued his functions without being any more sensible of the poison. We see by this that the effects of the poison of the manchinelle are different from those of the fish at Guadaloupe.. CVI. Abstract of a Letter from Mr. Wm. Arderon, F. R. S. to Mr. Henry Baker, F. R. S. on giving Magnetism and Polarity to Brass. p. 774. Mr. A. having made experiments on the magnetism of brass, among many pieces that he had tried, were several that readily attracted the needle; but whether they had this property originally, or received it by hammering, filing, clipping, or any other such like cause, he could not determine. He had a very handsome compass-box made of pure brass, as far as he could judge: the needle being taken out, and placed on a pin fixed properly in a board, and clear of all other magnetics, the box will attract this needle at half an inch distance; and if suffered to touch, will draw it full 90 degrees from the north or south points; and he thought those parts of the box marked north and south attracted the strongest. The cover of the box also attracted the needle nearly as much as the box itself. . As to your supposition, says Mr. A., that iron may be mixed with the brass, I do not know; but I have been informed it cannot be, as brass fluxes with a much less degree of heat than iron, and iron naturally swims on fluid brass. Besides, many of the specimens of brass I have tried were new as they came from the mill, where they were wrought into plates, and I presume were not mixed ;* yet these I have given the magnetic virtue to when they had it not; and some pieces of brass, which naturally attract the needle, seem to the eye as fine a bright yellow as any other, and are as malleable as any I ever met with. Pieces of brass without any magnetic power, by properly hammering and giving them the double touch, after Mr. Mitchel's method, I have made attract and repel the needle, as a magnet does, having 2 regular poles. You will observe, when you try this bar, that the same poles repel each other, and the contrary poles attract; which proves this piece of brass to be endued with true magnetic virtue and po larity. However it must be noted, that though the same poles repel each other, *This refers to Mr. Baker's having supposed, that old iron and old brass may be mixed sometimes, and melted down together.-Orig. |
