2 By the method of Section 5, using H, and H,, we determine the perihelion distance and other elements of the parabola which would pass through those two points. The distance so determined is 72.7. It will be seen by Diagram C. how nearly it coincides with H, and H. The vertex of the parabola (i. e. the perihelion) is at P; its longitude (7) measuring 268° 55'. The time T may be obtained thus:--By means of the pediometric line N N' cut off Q, making the area subtended by HQ = that subtended by H, H2, the time being 9 days. It is easy to measure the small arc RP as 1.21 day. T therefore becomes June 9-18 +9 − 1.21 = June 16.97. The elements of the orbit are now ascertained : 1 If we take the case of the comet dealt with in the last example it will appear that the space due to the parabolic orbit between the dates June 9 and June 24 should bear the proportion of about 1.671:1. (viz. √1,016 × √2) to that traversed by the earth during the same period--a proportion which it will be found by measurement on the diagram has been very nearly obtained. 727 SECTION 9. To form an ephemeris of a Comet. If it be desired to form an ephemeris from the elements of a comet's orbit the procedure graphically would be as follows: Taking the case of Example 1, namely given the perihelion distance 0.636, and the date of perihelion passage Aug. 22-23; let it be desired to find the comet's place in R. A. and Decl. for Sept. 2.33. By dividing the normal value of daily motion at perihelion (Sect. 7) by 0.636 we obtain in this case in terms of our scale 3.0518, and for the subtended area 97-049. This requires for 11-I days an area of 1077.24. The formula given in Section 5, namely, A = ~(~+3a), would y (x suffice for finding the place on the orbit, but would require the solution of a cubic equation, and as that might be tedious, it would be more convenient to use the formula as a correction of a value otherwise obtained. By the pediometric method we should obtain 11.1 x 3.0518 33.875 for the ordinate. This however would be somewhat too great, as the space inclosed between the chord and the arc is too large to be neglected. But the excess can be easily calculated. : = From the equation to the parabola we readily obtain the quantity 4.5107 as the abscissa due to 33-875, and from the y (x+3a) we obtain A = 1105·25. formula A = 6 The area in excess, 28.01, is proportional to o-29 of a day, so that instead of the place due to Sept. 2.33 we have that of Sept. 2.52, which will probably answer the purpose aimed at nearly as well: if not, an adjustment could be easily made. If the point H, had been at a greater distance from the Perihelion, it would have been requisite to have approximated to it by stages by the pediometric method, as shown in Section 2, the place so obtained to be corrected by the formula used above. The point H, on the orbit having been obtained, draw through it the straight line HR, perpendicular to the node, and find upon it the point J where RJR H, cos i. Join E, J and this will give the longitude of the required place. gives the latitude. RJ tan i Also = tan / E. J The R. A. and Decl. may now either be computed or solved graphically. CHAPTER VII. A CATALOGUE OF ALL THE COMETS WHOSE ORBITS HAVE HITHERTO BEEN COMPUTED. WHEN a new comet has been discovered, the first thing to be done is to obtain 3 observations of it, whereby the elements of the orbit may be computed. The computer will then examine a catalogue of comets to see if he can identify the newly-found stranger with any that have been before observed". The value of a good catalogue is obvious; and therefore I have compiled as complete a one as possible. In the preparation of the following list, care has been taken that only the most reliable orbits that were to be obtained should be inserted, the general rule being to prefer the one which was derived from the longest arc, other things being satisfactory. Among the authorities consulted may be mentioned Pingré, Hussey, Olbers, Cooper, Hind, Arago, Galle, and many others. The Epoch of perihelion passage is expressed in Greenwich Mean Time, N.S., since 1582. The Longitudes of Perihelion and of the Ascending Node are given for the respective epochs, but for any other epoch an allowance must be made for the effect of precession. This allowance is additive for subsequent dates and subtractive for previous ones, as follows: 1 year = 50"; 100 years = 1° 23′ 46′′; 1000 years = 13° 56′ 50′′. The periods assigned in the column of "Duration of Visibility" are subject to much uncertainty, more especially in the case of the ancient comets. a In the Annuaire de l'Observatoire Royal de Bruxelles, 1883, at p. 70, there will be found a catalogue of comets arranged in the order of the Inclinations. 1. It is said to have separated into two parts. 3. It had a short but brilliant tail. 4. An apparition of Halley's comet (), mentioned by Dion Cassius as having been suspended over Rome previous to the death of Agrippa. 5. An apparition of Halley's comet (?). It had a tail 8° long. 6. An apparition of Hulley's comet. 9. Elements somewhat doubtful. It had a tail 30° long. 11. Undoubtedly an apparition of Halley's comet. 12. It had a tail 10 feet long!! 13. A mean orbit. It had a tail 10° long. 14. Elements very reliable. On Sept. 8 it had a tail 40° long. 15. Elements very uncertain. 16. An apparition of Halley's comet. 17. It had a tail about 30° long. 18. Tolerably trustworthy. The maximum length of the tail was 80°, but it dwindled down to 30° in a fortnight. 20. Probably an apparition of Halley's comet. Mentioned by several Saxon writers. 21. These elements appear to have escaped the notice of recent cometographers, though given by Pingré; but has it been confounded with the following? 22. Possibly an apparition of Halley's comet. This is the famous object which created such universal dread throughout Europe in 1066. In England it was looked upon as a presage of the success of the Norman invasion. 23. Elements satisfactory. 24. A tail 50° long was seen in China, and much bifurcated. |