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ther the intereft I feel in this fubject warrants fuch an indulgence, I know not; but this I know, that I feel, and that I glory in feeling warm upon the prefent occafion! I fhould be the most bafe, the most ungrateful, and the moft defpicable of mankind if I were infenfible to the warmest fentiments of animation in the business fo interesting to a body of men, to whom I am bound in every tie of gratitude; but the wonder is not that I, or a particular part of the House, fhould feel warm upon a fubject in which the deareft, the unalienable, the most invaluable rights of the whole body of the electors of Great-Britain are deeply involved; the wonder is, that there fhould not be an
universal glow, an emulation for zeal in every member of this Houfe! If the Houfe difapprove of what I have faid, they may cenfure me: they cannot make me retract it. I fhall continue to infift on the truth and propriety of what I have faid: I will not retract one fyllable, one letter of it; nor even one thade of an idea on the fubject: but, if Mr. Murphy is in the House, and I fhall not be forry that he is, I will repeat it to him, in that, or any other fhape in which I may affert the propriety of the question."
Mr. Pitt faid a few words, upon the eloquent invectives he said he had juft heard; and then defired Mr. Murphy to be called in.
ON THE MEANS OF DISCOVERING THE DISTANCE, MAGNITUDE, &c. OF THE FIXED STARS, IN CONSEQUENCE OF THE DIMINUTION OF THE VELOCITY OF THEIR LIGHT, IN CASE SUCH A DIMINUTION SHOULD be founD TO TAKE PLACE IN ANY OF THEM, AND SUCH OTHER DATA SHOULD BE PROCURED FROM OBSERVATIONS, AS WOULD BE FARTHER NECESSARY FOR THAT PURPOSE. BY THE REV. JOHN MICHELL, B. D. F. R. S. IN A LETTER TO HENRY CAVENDISH, ESQ. F. R. S. AND A. S.
Read November 27, 1783.
HE method, which I mentioned
don, by which it might perhaps be poffible to find the diftance, magnitude, and weight of fome of the fixed ftars, by means of the diminution of the velocity of their light, occurred to me foon after I wrote what is mentioned by Dr. Priestley in his Hiftory of Optics, concerning the diminution of the velocity of light in confequence of the attraction of the fun; but the extreme difficulty, and perhaps impoffibility, of procuring the other datà neceffary for this purpofe appeared to me to be fuch objections against the fcheme, when I first thought of it, that I gave it then no farther confideration. As fome late obfervations, however, begin to give us a little more chance of procuring fome at least of LOND. MAG. April 1785.
Thornhill, May 26, 1783. thefe data, I thought it would not be
prized of the method I propofe (which, as far as I know, has not been fuggefted by any one elfe) left, for want of being aware of the ufe which may be made of them, they should neglect to make the proper obfervations when in their power; I fhall therefore beg the favour of you to prefent the following paper on this fubject to the Royal Society. I am, &c.
THE very great number of ftars that have been discovered to be double, triple, &c. particularly by Mr. Herfchel*, if we apply the doctrine of chances, as I have heretofore done .in my Enquiry into the probable Parallax, &c. of the Fixed Stars," publifhed in the Philofophical Tranfactions for the year 1767, cannot leave a doubt with
*See his Catalogue of Stars of this kind, published in the Philofophical Tranfactions for the year 1782, which is indeed a moft valuable prefent to the aftronomical world. By a happy application of very high magnifying powers to his telescopes, and by a most perfevering induftry in ob ferving, he has made a very wonderful progrefs in this branch of aftronomy, in which almoft nothing of any confequence had been done by any one before him,
with any one, who is properly aware of the force of thofe arguments, that by far the greateft part, if not all of them, are fyftems of ftars fo near to each other, as probably to be liable to be affected fenfibly by their mutual gravitation; and it is therefore not unlikely, that the periods of the revolutions of fome of these about their principals (the fmaller ones being, upon this hypothefis, to be confidered as fatellites to the other) may fome time or other be discovered.
2. Now the apparent diameter of any central body, round which any other body revolves, together with their apparent diftance from each other, and the periodical time of the revolv ing body being given, the denfity of the central body will be given likewife. See Sir Ifaac Newton's Prin. B. III. Pr. viii. Cor. 1.
3. But the denfity of any central body being given, and the velocity any other body would acquire by falling towards it from an infinite height, or, which is the fame thing, the velocity of a comet revolving in a parabolic orbit, at its furface, being given, the quantity of matter, and confequently the real magnitude of the central body, would be given likewise.
4. Let us now fuppofe the particles of light to be attracted in the fame manner as all other bodies with which we are acquainted; that is, by forces bearing the fame proportion to their vis inertia, of which there can be no reafonable doubt, gravitation being, as far as we know, or have any reafon to believe, an univerfal law of nature. Upon this fuppofition then, if any one of the fixed ftars, whofe denfity was known by the above-mentioned means, fhould be large enough fenfibly to affect the velocity of the light iffaing from it, we fhould have the means of knowing its real magnitude, &c.
5. It has been demonftrated by Sir Ifaac Newton, in the 39th propofition of the first book of his Principia, that if a right line be drawn, in the direction of which a body is urged by any forces whatfoever, and there be erected at right angles to that line perpendiculars every where proportional to
the forces at the points, at which they are erected refpectively, the velocity acquired by a body beginning to move from reft, in confequence of being fo urged, will always be proportional to the fquare root of the area described by the aforefaid perpendiculars. And hence,
6. If fuch a body, instead of beginning to move from reft, had already fome velocity in the direction of the fame line, when it began to be urged by the aforefaid forces, its velocity would then be always proportional to the fquare root of the fum or difference of the aforefaid area, and another area, whofe fquare root would be proportional to the velocity which the body had before it began to be fo urged; that is, to the fquare root of the fum of thofe areas, if the motion acquired was in the fame direction as the former motion, and the fquare root of the difference, if it was in a contrary direction. See Cor. 2. to the abovefaid propofition.
7. In order to find, by the foregoing propofition, the velocity which a body would acquire by falling towards any other central body, according to the common law of gravity, let C in the figure, reprefent the centre of the central body, towards which the falling body is urged, and let CA be a line drawn from the point C, extending infinitely towards A. If then the line RD be fuppofed to represent the force, by which the falling body would be urged at any point D, the velocity which it would have acquired by falling from an infinite height to the place D, would be the fame as that which it would acquire by falling from D to C with the force RD, the area of the infinitely extended hyperbolic space ADRB, where RD is always inverfely proportional to the fquare of DC, being equal to the rectangle RC contained between the lines RD and CD. From hence we may draw the following corollaries.
8. Cor. 1. The central body DEF remaining the fame, and confequently the forces at the fame diftances remaining the fame likewife, the areas of the rectangles RC, C will always be in
verfely as the diftances of the points D, d from C, their fides RD, rd being inverfely in the duplicate ratio of the fides CD, Cd: and, therefore, because the velocity of a body falling from an infinite height towards the point C, is always in the fub-duplicate ratio of thefe rectangles, it will be in the fubduplicate ratio of the lines CD, Cd inverfely. Accordingly the velocities of comets revolving in parabolic orbits are always in the fub-duplicate ratio of their diftances from the fun inverfely; and the velocities of the planets, at their mean diftances (being always in a given ratio to the velocity of fuch comets, viz. in the fub-duplicate ratio of 1 to 2) must neceffarily obferve the fame law likewife. 9. Cor. 2. The magnitude of the central body remaining the fame, the velocity of a body falling towards it from an infinite height will always be, at the fame diftance from the point C, taken any where without the central body, in the fub-duplicate ratio of its denfity; for in this cafe the diftance Cd will remain the fame, the line rd only being increafed or diminished in the proportion of the denfity, and the rectangle rC confequently increased or diminished in the fame proportion.
10. Cor. 3. The denfity of the -central body remaining the fame, the velocity of a body falling towards it from an infinite height will always be as its femi-diameter, when it arrives at the fame proportional diftance from the point C; for the weights, at the furfaces of different fphæres of the fame denfity are as their refpective femidiameters; and therefore the fides RD and CD, or any other fides rd and Cd, which are in a given ratio to thofe femi-diameters, being both increafed or diminished in the fame proportion, the rectangles RC or C will be increafed or diminished in the duplicate ratio of the femi-diameter CD, and confequently the velocity in the fimple ratio of CD.
11. Cor. 4. If the velocity of a body falling from an infinite height towards different central bodies is the fame, when it arrives at their furfaces, the denfity of those central bodies muft
be in the duplicate ratio of their femidiameters inverfely; for, by the laft cor. the denfity of the central body remaining the fame, the rectangle RC will be in the duplicate ratio of CD; in order, therefore, that the rectangle RC may always remain the fame, the line RD must be inverfely, as CD, and confequently the denfity inversely, as the fquare of CD.
12. Cor. 5. Hence the quantity of matter contained in those bodies must be in the fimple ratio of their femidiameters directly; for the quantity of matter being always in a ratio compounded of the fimple ratio of the denfity, and the triplicate ratio of their femi-diameters, if the density is in the inverfe duplicate ratio of the femidiameters, this will become the direct triplicate and inverse duplicate, that is, when the two are compounded together, the fimple ratio of the femidiameters.
13. The velocity a body would. quire by falling from an infinite height towards the fun, when it arrived at his furface, being, as has been faid before in article 3d, the fame with that of a comet revolving in a parabolic orbit in the fame place, would be about 20,72 times greater than that of the earth in its orbit at its mean distance from the fun; for the mean distance of the earth from the fun, being about 214,64 of the fun's femi-diameters, the velocity of fuch a comet would be greater at that distance than at the diftance of the earth from the fun, in the fub-duplicate ratio of 214,64 to 1, and the velocity of the comet being likewife greater than that of planets, at their mean diftances, in the subduplicate ratio of 2 to 1; these, when taken together, will make the subduplicate ratio of 429,28 to 1, and the fquare root of 429,28 is 20,72, very nearly.
14. The fame refult would have been obtained by taking the line RD proportional to the force of gravity at the fun's furface, and DC equal to his femi-diameter, and from thence computing a velocity, which should be proportional to the fquare root of the area RC when compared with the square Iia
root of another area, one of whofe fides fhould be proportional to the force of gravity at the furface of the earth; and the other fhould be, for inftance, equal to 16 feet, 1 inch, the fpace a body would fall through in one fecond of time, in which cafe it would acquire a velocity of 32 feet, 2 inches per fecond. The velocity thus found compared with the velocity of the earth in its orbit, when computed from the fame elements, neceffarily gives the fame refult. I have made ufe of this latter method of computation upon a former occafion, as may be feen in Dr. Prieftley's Hiftory of Optics, p. 787, &c. but I have rather chofen to take the velocity from that of a comet, in the article above, on account of its greater fimplicity, and its more immediate connexion with the fubject of this paper.
15. The velocity of light, exceeding that of the earth in its orbit, when at its mean distance from the fun, in the proportion of about 10.310 to 1, if we divide 10.310 by 20,72, the quotient 497, in round numbers, will exprefs the number of times, which the velocity of light exceeds the velocity a body could acquire by falling from an infinite height towards the fun, when it arrived at his furface; and an area whofe fquare root fhould exceed the fquare root of the area RC, where RD is fuppofed to reprefent the force of gravity at the furface of the fun, and CD is equal to his femi-diameter, in the fame proportion, muft confequently exceed the area RC in the proportion of 247.009, the fquare of 497 to 1.
16. Hence, according to article 10, if the femi-diameter of a fphere of the fame denfity with the fun were to exceed that of the fun in the proportion of 500 to 1, a body falling from an infinite height towards it, would have acquired at its furface a greater velocity than that of light, and confequently, fuppofing light to be attracted by the fame force in proportion to its vis inertia, with other bodies, all light emitted from fuch a body would be made to return towards it, by its own proper gravity.
17. But if the femi-diameter of a fphære, of the fame denfity with the fun, was of any other fize less than 497 times that of the fun, though the velocity of the light emitted from fuch a body, would never be wholly destroyed, yet would it always fuffer fome diminution, more or lefs, according to the magnitude of the faid fphære; and the quantity of this diminution may be eafily found in the following manner: fuppofe S to reprefent the femi-diameter of the fun, and aS to reprefent the femi-diameter of the propofed fphære; then, as appears from what has been fhewn before, the fquare root of the difference between the fquare of 497 S and the fquare of aS will be always proportional to the ultimately remaining velocity, after it has fuffered all the diminution it can poffibly fuffer from this caufe; and confequently the difference between the whole velocity of light, and the remaining velocity, as found above, will be the diminution of its velocity. And hence the diminution of the velocity of light emitted from the fun, on account of its gravitation towards that body, will be fomewhat lefs than a 494.000dth part of the velocity which it would have had if no fuch diminution had taken place; for the fquare of 497 being 247.009, and the fquare of 1 being 1, the diminution of the velocity will be the difference between the fquare root of 247.009, and the fquare root of 247.008, which amounts, as above, to fomewhat lefs than one 494.000th part of the whole quantity.
18. The fame effects would likewife take place, according to article 11, if the femi-diameters were different from those mentioned in the two laft articles, provided the denfity was greater or lefs in the duplicate ratio of thofe femi-diameters inversely.
19. The better to illuftrate this matter, it may not be amifs to take a particular example. Let us fuppofe then, that it fhould appear from obfervations made upon fome one of thofe double ftars above alluded to, that one of the two performed its revolution round the other in 64 years, and that the central one was of the fame den
fity with the fun, which it must be, if its apparent diameter, when feen from the other body, was the fame as the apparent diameter of the fun would be if feen from a planet revolving round him in the fame period: let us further fuppofe, that the velocity of the light of the central body was found to be less than that of the fun, or other ftars whofe magnitude was not fufficient to affect it fenfibly, in the portion of 19 to 20. In this cafe then, according to article 17, the fquare root of 247.009 SS must be to the fquare root of the difference between 247.009 SS and aaSS as 20 to 19. But the fquares of 20 and 19 being 400 and 361, the quantity 247.009 SS muft therefore be to the difference between this quantity and aaSS in the fame proportion, that is as 247.009 to 222.925,62; and aaSS muft confequently be equal to 24.083,38 SS, whofe fquare root 155,2 S nearly, or, in round numbers, 155 times the diameter of the fun, will be the diameter of the central ftar fought.
20. As the fquares of the periodical times of bodies, revolving round a central body, are always proportional to the cubes of their mean diftances, the distance of the two bodies from each other muft therefore, upon the foregoing fuppofitions, be fixteen times greater in proportion to the diameter of the central body, than the distance of the earth from the fun in proportion to his diameter; and that diameter being already found to be alfo greater than that of the fun in the proportion of 155,2 to 1, this diftance will confequently be greater than that of the earth and fun from each other in the proportion of 16 times 155,2, that is 2483,2 to 1.
21. Let us farther fuppofe, that from the obfervations, the greatest diftance of the two ftars in queftion appeared to be only one fecond; we must then multiply the number 2483,2 by 206.264,8, the number of feconds in
the radius of a circle, and the product 512.196.750 will fhew the number of times which fuch a ftar's diftance from us must exceed that of the fun. The quantity of matter contained in fuch a
ftar would be 155,2 or 3.738.308 times as much as that contained in the fun; its light, fuppofing the fun's light to take up 87.7" in coming to the earth, would, with its common velocity, require 7.900 years to arrive at us, and 395 years more on account of the diminution of that velocity; and fuppofing fuch a ftar to be equally luminous with the fun, it would still be very fufficiently vifible, I apprehend, to the naked eye, notwithstanding its immenfe diftance.
22. In the elements which I have employed in the above computations, I have fuppofed the diameter of the central ftar to have been obferved, in order to afcertain its density, which cannot be known without it; but the diameter of fuch a ftar is much too fmall to be obferved by any telescopes yet exifting, or any that it is probably in the power of human abilities to make; for the apparent diameter of the central ftar, if of the fame denfity with the fun, when feen from another body, which would revolve round it in 64 years, would be only the 1717th part of the distance of those bodies from each other, as will appear from multiplying 107,32, the number of times the fun's diameter is contained in his diftance from the earth, by 16, the greater proportional diftance of the revolving body, correfponding to 64 years instead of 1. Now the 1717th part of a fecond must be magnified 309.060 times in order to give it an apparent diameter of three minutes; and three minutes, if the telescopes were mathematically perfect, and there was no want of diftinctness in the air, would be but a very small matter to judge of*.
23. But though there is not the leaft
* In Mr. Herfchel's Obfervations upon the Fixed Stars abovementioned, almoft all of them are reprefented as appearing with a well-defined round difc. That this is not the real dife, but only an optical appearance, occafioned perhaps by the conftitution of the eye, when the pencil, by which objects are feen, is fo exceedingly fmall as thofe which he employed upon this occafion, is very manifeft, from the obfervations themselves, of which indeed Mr. Herfchel feems to be himself suffi