Prob. x. To find the [prefent] value of a given fum, payable on the extinction of either of two certain lives, hould either of these lives be the last that fails of three given lives.

The folutions to all thefe problems are given from the real probabilities of life, derived from bills of mortality, and not from M. De Moivre's hypothefis of equal decrements; on which, as Mr. Morgan had fhewn in his former papers, we ought not to depend. He adds,

I have now given general rules for determining the value of reverfions depending upon three lives in every cafe which, as far as I can difcover, will admit of an exact folution. The remaining cafes, which are nearly equal in number to thofe I have inveftigated, involve a contingency, for which it appears very difficult to find fuch a general expreffion, as fhall not render the rules much too complicated and laborious. The contingency to which I refer is that of one life's failing after another, in any given time. The fractions expreffing this probability are every year increafing, fo that the value of the reverfion must be reprefented by as many feries, at least, as are equal to the difference between the age of one of the lives, and that of the oldeft life in the table of oblervations. I have indeed fo far fucceeded in the method of approximation, as that the reverfion may be generally afcertained within about one-fiftieth part of its exact value; but I fhall not trouble the Royal Society at prefent with these investigations.'

We fhall juft remark, that there appears to be fome difagreement between the enunciation and folution of the tenth problem. We fhall give both in Mr. Morgan's own words:

QUEST. To find the value of a given fum, payable on the decease of B or C, fhould either of them be the laft that fails of the three lives, A, B, and C. In the folution, Mr. Morgan fays, The fum S can be received in the first year only on the extinction of the three lives, restrained to the contingency of A's life having been the firft or fecond that failed:" but it appears to us that, according to the enunciation of the problem, the fum S is receiveable in the first year, if A, and either B, or C, die in it, though the other does not. Abstract of a Regifter of the Barometer, Thermometer, and Rain, at Lyndon, in Rutland, by Thomas Barker, Efq; with the Rain in Surrey and Hampshire; for the Year 1790.

This, we believe, is the twentieth regifter of the kind that Mr. Barker has fent regularly in every year; and we most fincerely with him health and fpirits to continue them for many years to come. The register of the rain in Surrey was kept at South Lambeth; and thofe for Hampshire at Selbowin and Fyfield. To his ufual general account of the feafons, Mr. Barker has fubjoined an account of places where chalk

has

has been lately found, and where it was not known to exist before.

Defeription of a fimple Micrometer for measuring (mall Angles with the Telescope. By Mr. Tiberius Cavallo, F. R. S.

This micrometer confifts of a thin and narrow flip of mother-of-pearl, finely divided, and fituated in the focus of the eve-glafs of a telefcope, juft where the image of the object is formed. It is immaterial whether the telefcope be a refractor or a reflector, provided the eye-glafs be a convex lens, and not a concave one, as in the Gallilean conftruction.

The fimpleft way of fixing it, is to flick it on the diaphragm, which generally stands within the tube, and in the focus of the eye glas. When thus fixed, if the obferver look through the eye-glafs, the divifions of the micrometrical fcale will appear very diftin, unless the diaphragm be not exactly in the focus; in which cafe the micrometrical fcale must be placed exactly in the focus of the eye-glafs, either by pufhing the diaphragm backward or forward, when that is practicable; or elfe the fcale may be easily removed from one or the other surface of the diaphragm, by the interpofition of a circular piece of paper or card, or by a bit of wax. This conftruction is fully fufficient, when the telescope is always to be ufed by the fame perfon: but when different perfons are to use it, then the diaphragm, which fupports the micrometer, must be constructed fo as to be eafily moved backward or forward, though that motion needs not be greater than about a tenth or an eighth of an inch. This is neceffary, because the distance of the focus of the fame lens appears different to the eyes of different perfons; and therefore, whoever is to use the telescope for the menfuration of any angle, muft first unfcrew the tube, which contains the eye-glafs and micrometer, from the rest of the telefcope; and, looking through the eye-glafs, muft place the micrometer where the divifions of it may appear quite diftin&t to his eye.

A new Method of invefligating the Sums of Infinite Series. By the Rev. Samuel Vince, A. M. F. R. S.

The fummation of infinite feries is a fubject, not only of curious fpeculation, but also of the greatest importance in various branches of the mathematics and natural philofophy; in confequence of which it has always claimed a very confiderable fhare of attention from the most celebrated mathematicians. For these reasons, Mr. Vince does not think it necef fary to make any apology for offering this paper to the public, which contains a new and very expeditious method of obtain

ing the fums of a great variety of feries, moft of which have never before been treated.

As the method of fummation here pointed out, depends on the fums of the reciprocals of the powers of the natural numbers, tables of fuch fums are prefixed, extending as far as the 40th power, to twelve places of decimals, by which the fums of the feries will be found true to ten or eleven places; and if greater accuracy be required, which feldom happens, it may be easily obtained by continuing the tables.

The method is illuftrated by a great number of examples, which render the ufe of the tables very easy.

This part concludes with the ufual lift of presents to the Society, index to the volume, &c.

Wa..s.

ART. X. Tranfactions of the Royal Irish Academy, MDCCLXXXIX. 18s. Boards. Elmiley. 1791.

Vol. III.

4to.

CLASS of SCIENCE.

Experiments on the Alkaline Subftances used in Bleaching, and on the Colouring Matter of Linen Yarn. By Richard Kirwan, Efq. F. R. S. &c.

IN thefe experiments, if we may take the author's word for

it, there is not much ingenuity: but there is certainly much. well-directed labour, and much valuable information to those who are concerned in the objects of them; and though they were immediately undertaken for the ufe of the bleachers of our fifter kingdom, at a time when a fcarcity of foreign potashes prevailed, and when it was feriously queftioned whether any of home manufacture could be made to anfwer the refpective purposes to which rote practice had applied the foreign forts, yet the better informed bleachers of our own country will profit by them; and there are other kinds of bufinefs in which a knowlege of the relative ftrengths of the different alcaline fubftances to be found in commerce, and an eafy method of afcertaining the ftrength and quality of any given one, cannot fail to be interefting.

Mr. Kirwan begins with a complete analyfis of barilha, (we follow his own orthography,) inveftigating, with minute precifion, the quantity of fixed air, mineral alcali, common falt, Glauber's falt, charcoal, calcareous, muriatic, argillaceous, and filiceous earths contained in it: but being difgufted, as well he might, by the tedioufnefs of thefe operations, and confidering that the pure alcali is the only ingredient in these subftances that is ufeful in bleaching, he contrived a fimple and

practicable

practicable method of determining the quantity of that principle, wherever it exists in an active or uncombined ftate. Solution of alum is the teft; and the quantity of alcali is in proportion to the quantity of alum earth which it precipitates.

Mr. K. boils an ounce of the given alcaline substance in half a pint or a pint (according as it contains lefs or more earth,) of pure water, in a Florence fiafk, for a quarter of an hour, and, when cold, filters the folution. This liquor is again heated, and a faturated folution of alum, likewife made hot, is poured into it at intervals, fo long as any precipitation enfues, or till the liquor changes the colour of fyrup of violets, or blue litmus'd paper, to red. The whole is then poured into a filter, and the precipitated earth is washed with hot water till the water paffes through taftelefs. The filter, with the earth. on it, is dried, till they feparate cafily; and the earth is farther dried in a cup of Staffordshire ware or hot fand, till it ceases to stick to glafs or iron: it is then reduced to powder in the cup, with a glafs peftle, and kept for a quarter of an hour' in a heat from 470 to 500°.

By this heat, the fixed air, which the alum earth may have taken up in the precipitation, will in a great measure be expelled but, for the complete feparation of it, recourfe is had to marine acid. The dried earth is put into a Florence flask and weighed about an ounce of fpirit of falt, in another flask, is placed in the fame fcale; and both of them are counterbalanced in the oppofite fcale. The acid being then poured on the earth, an effervefcence generally enfues, and this fcale becomes lighter the weight neceflary to be added for restoring the equilibrium, will be the weight of the air expelled by the acid, and must be fubtracted from that of the dried earth.

The mineral and vegetable alcali are known to differ greatly from one another in their power of faturating acids, and of precipitating fubftances diffolved in them. Mr. Kirwan finds that an ounce Troy, (or 480 grains,) of pure mineral alcali, precipitates 725 grains of earth of alum: but the same quantity of the vegetable alcali precipitates only 331. The quantity of alum earth, therefore, precipitable by an ounce of any alcaline fubftance of either clafs, being known, it will be eafy to compute, from thefe refpective ftandards, the quantity of pure alcali contained in it.

Crystallized foda was found to contain only 96 grains on the ounce, or, which is the fame thing, 20 pounds on Ice, of pure mineral alcali; all the reft of it being water and fixed air. In fweet barilha, the alcali amounts to 24 on 100; fo that this fubftance, though nearly one half of it confifts of earthy matters indiffoluble in water, is more ftrongly alcaline than the

pure

pure cryftals. In the best kelps of home manufacture, the alcali fell fhort of 5 on 100.

In 100 parts of the Dantzick pearl-afh, the vegetable alcali amounted to fomewhat above 63; and in the cafhup afhes, to above 19; in the common Irish weed-afhes, to little more than 1; and in the fame afhes, when calcined, to a little more than 45.

The author next confiders the different methods of procuring the two fpecies of alcaline falts, and gives fome general directions for the manufacture of pot and pearl afhes: but on thefe fubjects we meet with little more than what is pretty generally known, (though not to those for whom he writes,) and there are fome pofitions which we could wish he had afcertained by experiment. His pot-afbes are prepared by evaporating a lye of vegetable afhes to drynefs in iron pots; and pearlabes, by calcining this dry falt to whitenefs. In this laft operation, he fays, particular care fhould be taken that it should not melt, as the extractive matter would not be thoroughly confumed, and the alcali would form fuch a union with the earthy parts as could not eafily be diffolved.' He has added this caution, as Dr. Lewis and Mr. Doffie have inadvertently directed the contrary.' We apprehend that here is a little inaccuracy, and that it was not for pearl-afh, but for the unrefined pot-afb, that these gentlemen directed fufion. The fact is, that the American pot-alhes, examined by them, had unqueftionably fuffered fufion; which was effected in the fame iron pot in which the evaporation was finished, by rather increasing the fire at the end of the procefs: by this management, one of the most troublesome operations in the whole manufacture, the feparation of the hard falt from the veffels with hammers and chisels, was avoided; and though the extractive matter was not confumed, it was burnt to an indiffoluble coal; fo that the falt, though black itself, produced a pale or colourless folution, and was uncommonly ftrong.

Mr. Kirwan has given alfo tables of the quantities of afhes and falt obtained from different vegetables; and he concludes from them, 1. That, in general, weeds yield much more ashes, and their ashes much more falt, than woods; and that, confequently, as to falts of the vegetable alcali kind, neither America, Triefte, nor the northern countries, poffefs any advantage over us.-2dly, That of all weeds, fumitary produces most falt, and next to it wormwood: but if we attend only to the quantity of falt in a given weight of afhes, the afhes of wormwood contain moft. Trifolium fibrinum also produces more ashes and falt than fern.'

From the curious experiments on the colouring matter of linen yarn, which conclude the paper, it appears, that this REV. MAY 1792.

F

fubftance