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ray of light might be refracted backwards and forwards by different mediums, as water, glafs, &c. provided it was fo done, that the emergent ray fhould be parallel to the incident one, it would ever after be white; and conversely, if it fhould come out inclined to the incident, it would diverge, and ever after be coloured. From which it was natural to infer, that all spherical object glaffes of telescopes must be equally affected by the different refrangibility of light, in proportion to their apertures, whatever material they < may be formed of.

But it seems worthy of confideration, that notwithstanding this notion has been generally adopted as an incontes<tible truth, yet it does not seem to have been hitherto fo confirmed by evident experiment, as the nature of fo important a matter juftly demands; and this it was that determined me to attempt putting the thing to iffue by experiment."

If we mistake not, opticians did not think that the different refrangibility of light was in proportion to the apertures; but rather that it was owing to the deviation from the true figure which the object-glafs fhould have to refract all the rays to the fame focus; and thofe rays which did fall short of, or furpaffed the focus, caufed the colouring round the edge of the image. Befides it is well known, that the fmaller the part of the fpheric surface is, in respect to the radius, the less colour is to be feen through the image; and it is for this reason, that refracters are made fo long, in order to get a larger field with lefs error from the true figure.

After this the Author enumerates feveral experiments he made in a glafs prifmatic veffel filled with water, with a glafs prifm in it; but as this is the fame with the eighth experiment of Sir Ifaac Newton's optics, book I. part ii. after prop. 8; it would be needlefs to repeat here, any more of it, than that the refult was quite contrary to the prefent: for the object, though not at all refracted, was yet as much infected with prifmatic colours, as if it had been feen through a glafs prifm, whose refracting angle was near thirty degrees. From whence he concludes, that the divergency of the colours, by different fubftances, was by no means in proportion to the refractions; and, that there was a poffibility of refraction, without any divergency of the light at all.

In confequence of thefe experiments, he made fome object glaffes for telescopes, of two fpheric forms with water between them; which, he fays, were free from the errors ari

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fing

fing from the different refrangibility of light: but not fo diftinct as might have been expected, because the radii of the ⚫ spherical furfaces of thofe glaffes were required fo fhort, in order to make the refractions in the required proportions, that they must produce as great, or greater errors in the image, than thofe from the different refrangibility of • light.

As thefe experiments clearly proved, that different fub• ftances diverged the light very differently, in proportion to ⚫ the refraction; I began to fufpect, that fuch variety might • poffibly be found in different forts of glaffes, efpecially, as experience had already fhewn, that fome made much better object glaffes in the ufual way, than others: and as no fa⚫tisfactory cause had as yet been affigned for fuch difference, ⚫ there was great reafon to prefume, that it might be owing to the different divergency of the light by their refraction.

• I discovered a difference, far beyond my hopes, in the refractive qualities of different kinds of glafs, with refpect to ⚫ their divergency of colours. The yellow, or ftraw-colour'ed foreign fort, commonly called, Venice glass, and the English crown glafs, are very near alike in that refpect, though, in general, the crown glafs feems to diverge the light, rather the leaft of the two. The common plate ⚫ glass made in England diverges more; and the white crystal, ⚫ or flint English glafs, as it is called, the most of all.'

Our author made feveral trials, in order to find two forts of glafs whofe difference was the greateft, which were the crown glass, and the white flint, or cryftal; and he found, that when the refraction of the white glafs, was to that of the crown glass, as two to three, the refracted light was intirely free from colours. Whence of two fpherical glaffes which refract the light in contrary directions, the one must be concave, and the other convex; and as the rays are to converge to a real focus, the excefs of refraction must be in the convex, which therefore, muft be made of crown glafs, and the concave with white flint glafs and as the refractions of fpherical glaffes are in an inverfe ratio to their focal diftances, it is easy to make these distances in the ratio given above. But it must be remembered, that the fpheric glaffes must have as large radii as they will admit of, although their focal diftances are limitted. He obferves, that the refracting powers of the fame fort of glass, made at different times, vary; and that the two glaffes must be placed truly, on the common axis of the telescope, otherwife the defired effect will be in great measure destroyed.

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Article

Article 85. A further attempt to facilitate the refolution of Isoperi

m.trical Problems. By Thomas Simpfon, F. R. S.

This branch of the mathematics, is of an old date, as we find by Apollonius Pergiae; but it received its greatest improvement from the method of fluxions: for general rules are from thence formed, for all problems of that kind, which could not have been done by any other method. But as the progress of arts and fciences is gradual, moft writers upon fluxions, have added new problems to thofe already known: especially our learned author, who has treated this branch very copiouily.

He takes notice, in this paper, that about three years before, he had laid before the Royal Society, the investigation of a general rule, for the refolution of Ifoperimetrical problems, wherein only one of the two indeterminate quantities enters along with the fluxions into the expreffion, and which rale determines the greatest figures under given bounds; lines of the iwiteit defcent; folids of the leaft refiftances; with innumerable ether cafes. But as others may be proposed, fuch as actually may arife in inquiries into nature, wherein both the flowing quantities together with their fluxions, are jointly concerned; it is the investigation of a rule for the refolution of thefe, which he attempts in this paper.

In order to this a general propofition is laid down, the purport of which is as follows: Let 2, R, &c. reprefent variablę quantities, expreffed in terms of x and y, with proper co-efficients, and let q, r, &c. denote as many others expreffed in terms of x and y; then it is proposed to find an equation for the relation of x and y, fo that the fluent of Q.q+Rr+&c. correfpondding to a given value of x (or y,) may be a maximum, or mininum.

To folve this problem, he denotes the fluxions of 2 and R, by Q, Ry, and the fluxions of q and r, by qÿ, rÿ; and after fuppofing y and y alone variable, and the two extreme ordinates conflant, he finds flux 27+R1r&c. =q'Q+r1R, &c. for his final equation That is, in words, the fluxion of the fluxion of 24+Rr, making j only variable, and divided by, is equal to the fluxion of the fame quantity Qq+Rr, by making only y variable, and divided by j. Whereas the author gives the following

GENERAL RULE. Take the fluxion of the given expreffion, whofe fluent is required to be a maximum, or minimum, making y alone

variable

• variable, and having divided by ÿ; let the quotient be denoted by v: then take again the fluxion of the fam: expreffion, making y alone variable, which divided by ÿ; and then this laft quotient will be.'

The author obferves that, when y is not found in the quantity given, v will then be =0; and confequently, the expreffion for equal to nothing alfo. But if y be abfent, then will, and confequently, the value of v = a conftant quantity: inftead of y and y, x and x may be made fucceffively variable. Morever, if the cafe to be refolved, should be confined to other reftrictions, befides that of the maximum, or minimum; fuch as, having a certain number of other fluents, at the fame time equal to given quantities, the fame method may ftill be applied, with equal advantage, provided all thefe expreffions are connected together with proper co

efficients.

To exemplify by a particular cafe, the method of operation, he proposes the fluxionary quantity хпутур wherein

the relation of x and y is fo required, that the fluent, correfponding to the given value of x and y, fhall be a maximum, or minimum: and proceeding according to the foregoing rule, he finds p m+p p=1 p-n

m+p

p =p-n-i Xx p

required equation, by fuppofing

for the

I. As this equa

tion indicates that x and y increase together from o to infinite, when their exponents are both pofitives, or, that while y increases, decreafes, when the exponent of x is negative; it can by no means be concluded, that the fluent of the given expreffion, contains either a maximum, or minimum: unless fome other condition be annexed to make it fo, which is not mentioned.

He gives another example: That the fluent of 3" ym ż may be a maximum, or minimum, and that of xp yay to be equal to a given quantity. These two quantities joined together, with the indeterminate co-efficient b, gives xn yn x+ bxyy for the fum; and proceeding according to the rule, finds byq=mxp for the required equation. But as this equation is of the fame nature as the former, it cannot be concluded, that the propofed quantity contains a maximum, or minimum, which, we imagine, the author fhould have fhewn: for it is by no means fufficient to give

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an expreffion, and make its fluxion equal to nothing, in order to determine the maximumn, or minimum, without fhewing that it contains one in fact. The author himself, has fhewn feveral exceptions in his treatife upon fluxions, and yet he takes not the leaft notice of that, in this paper; on the contrary all his fuppofitions are vague, and by no means decifive.

Article 103. The invention of a general Method for determining the Sum of every 2d, 3d, 4th or 5th, &c. term of a Series, taken in order; the Sum of the whole Series being known. By Thomas Simpson, F. R. S.

The author fays, As the doctrine of feries is of very great ufe in the higher branches of the mathematics, and their application to nature, every attempt tending to extend that doctrine, may juftly merit fome degree of regard. The fubject of the paper, which I have now the honour to lay before the fociety, will be found an improvement of • fome confequence in that part of science. And how far the bufinefs of finding fluents may, in fome cases, be facilitated thereby, will appear from the examples fubjoined, in illuftration of the general method here delivered.'

He then proceeds to fhew by examples, how the method he propofes may be refolved; but we fhall only obferve, that the author's intention feems to have been, to fhew a different way of demonftrating Mr. Cote's theorems, in his Harmonia Menfurarum. It feems to be a favourite topic of the author's; for he has treated this fubject in a different manner, in several of his works; and in fome of them, extended it a great length: it has likewife been treated by many others (but by none fo elegantly as by the inventor himself), and continued by Dr. Smith, the publisher of this work. It is true, the general problem was given without a demonftration; because it depends on the divifion of the circle, and is therefore easily performed as Mr. Demoivre has fhewn in his Mifcellanea Analytica.

Article 110. Of the irregularities in the Motion of a Satellite, arif ing from the fpheroidical figure ofits primary Planet: in a letter to the Rev. James Bradley, D.D. Aftronomer Royal; F.R.S. and Member of the Royal Academy of Sciences at Paris; by Mr. Charles Walmefly, F. R. S. and Member of the Royal Academy of Sciences at Berlin, &c.

Since the time, (fays the author) that aftronomers have been enabled by the perfection of their inftruments to determine with great accuracy, the motions of the celestial

⚫ bodies,