on, 





of iU orbit If UM 



calculated from these 



tisfy all the subsequent observed places of UM oomet within 

 UM limiu of the errors of observation, or if the observation, can be 

 Misnod by any possible correction of the original elements, it may be 

 concluded that UM oomet revolves in an orbit which 1s sensibly para- 

 bolic. But if UM observed places exhibit a ijstematia deviation from 

 the corresponding result* assigned by UM parabolic elements, an indi- 

 cation is thereby afforded that the comet really moves hi an ellipse or 

 an hyperbola, and the orbit may be investigated de noro by means of 

 Uaus. s method. In this way the six elliptic elements of a comet may 

 be at ooo. obtained, by a process founded on merely three observed 

 positions of UM body. 



The earliest general method for computing the effects of planetary 

 perturbation on the movements of comets u due to Lagrange. The 

 process devised by that geometer is founded on the application of 

 mechanical quadratures to the theory of the variation of elements. 

 The orbit of UM comet is divided into a number of distinct sections, 

 and the influence of planetary perturbation upon the elements is sepa- 

 rately computed for the individual arcs. In applying this process to 

 each arc, fresh elements are obtained which are employed in the com- 

 putations of the arc immediately following. In regard to many comets, 

 the influence of planetary perturbation is sensible only in the vicinity 

 of the perihelion, and in such cases it is only there that the application 

 of the method becomes necessary. It may be remarked that Lagrange's 

 theory of comeUry perturbation has served as the basis of all subse- 

 quent researches on the subject 



Periodic Cvnutt. 



(1) Comett vhich hate returned to their perihelia ttaee the atabIM- 

 men! of Heir periodicity. 



HaUeffi Comet. An account has already been given of the circum- 

 stances connected with the perihelion passage of this famous oomet in 

 1759. The next return to the perihelion occured in the year 1835. 

 In 1812, the Academy of Sciences of Turin propoced the investigation 

 of its perturbations as the subject of a prize. Damolseau, an eminent 

 French geometer, was on this occasion the successful competitor. He 

 found that the comet would pass through the perihelion on the 4th of 

 November, 1885. 



In 1829 M. De PonWcoulant obtained the prize of the Academy of 

 Sciences of Paris, for his researches on the same subject. The result 

 of his first investigation indicated that the oomet would pass the peri- 

 helion on the 7th of November, 1835 ; but on subsequently taking 

 into account the disturbing action of the earth, and employing more 

 accurate values of the masses of the other disturbing bodies, he found 

 that the passage of the perihelion would take place on the 16th of 

 November. 



The perturbations of the comet on the occasion of the same peri- 

 helion passage, also formed the subject of elaborate investigations by 

 Rosonberger and Lehmann. two German mathematicians. Rosenberger 

 found that the comet would pass through the perihelion on the llth 

 of November ; according to Lehmann the passage would take place on 

 the 26th of the same month. 



It was expected by astronomers that the oomet would become visible 

 about the beginning of August This conjecture received a satisfactory 

 confirmation. The comet was first discovered on the 5th of August, 

 at the Observatory of Rome, by MM. Dumouohel and De Vico. 

 Towards the end of September it became visible to the naked eye. It 

 attained its greatest brilliancy about the middle of October. The 

 head then resembled a star of the second magnitude. The tail exhi- 

 bited an apparent length of about 20" in the countries of Northern 

 Europe, but in southern climates it was observed to extend > 

 are of 30. The comet was not much seen in the northern hemisphere 

 after the middle of November, having been shortly afterwards lost in 

 the sun's rays. Early in the following year it was seen at the Cape of 

 Good Hope, by Sir John Herschel and Mr. Maclear, and continued to 

 be observed till the 12th of May. 



The most complete investigation of the elements of the comet, 

 founded on all tho most trustworthy observations made in 1835-6, U 

 due to Westphalen, a German astronomer of great promise, who 

 shortly afterwards died at an early age. In order to exhibit the 

 accordance which existed between theory and observation in this 

 instance, we subjoin the elements of the comet as assigned respectively 

 by Ponteooulant and Westphalen. 



roaUcoulant. 

 Paa* of the peruwlion, \. 



M. T. Greenwich . . ) USi> " OT - " OI 

 Longitude of tin perihelion 804' IT 42" 

 LonitHade of tbo uccnding \ 4J , 1Q , j y , 



17' ' 48" 

 0-067J807 

 11-00001 



node 

 Inellnillon . 

 Exeentrldtr 

 Hsu distance 



I 



Wntphalcn. 

 1835, NOT. 15-03 

 304" SI' 33" 

 83 0' 69" 



17 45' 8" 

 00073909 

 17-99190 



this w*s also an apparition of Ualley'. comet The following are the 

 times of revolution corresponding to the passages of the perihelion 

 which have been deduced from the recorded observations : 



U has been already mentioned that the comet of 1456 was an 

 apparition of Halley's oomet The discovery of this fact is due to tin. 

 French astronomer Pingrd. It has been recently ascertained l>y tin 

 late M. Edward Biot, that a conspicuous oomet was observed in China 

 in the yew 1878, and M. Laugu-r has established beyond doubt that 



1ST! 1458 

 1450 liSl 



UM 1607 

 107 16SI 

 16MS 17J9 

 1749 183J 



Y, M. 



77-48 



75-11 



78-15 



74 9t 



70-49 



n , , 



The mean of these periods is 76-1 years. The deviation) from thin 

 result represent the effects of planetary perturbation. 



Enfie'i Comer. This comet, as in the case of the famous comet of 

 H alley, had been observed .on several occasions of its return to tho 

 wrihelion, before its periodicity was established. On the 26tli of 

 November, 1818, a comet was discovered by Pons at Marseil! 

 jnrnbolic elements of which were soon afterwards found to resemble 

 those of comets observed in 1805, 1705, and 1786. M. Encke was 

 induced by thn circumstance to investigate an elliptic 01 

 comets, and he found the tiiuo of revolution to be somewhat more than 

 1200 days. In 1822, on the occasion of its next return to the peri- 

 helion, it was rediscovered by M. Rumkor, at Paramatta, in New South 

 Wales. Upon tracing back its motion it was found to be in reality 

 Identical with the comets of 1805, 1705, and 1786, but a comparison of 

 the earlier with the more recent perihelion passages seemed to indicate 

 that the time of revolution was gradually becoming shorter. Professor 

 Encke was induced in consequence to suspect the existence of a re- 

 sisting modium, and adopting such an hypothesis, he calculated before- 

 hand the perihelion passage of 1825. The results were found to present 

 a satisfactory agreement with those derived from observation, and on 

 every subsequent occasion of the comet's return to the perihelion, its 

 motion has been successfully computed beforehand by Professor I 

 on the supposition of a resisting' medium. The following tai 

 tractcd from a paper by Professor Encke, exhibits the gradual shoi 

 of the time of revolution, M indicated by observation. This remark 

 does not apply to the perihelion passages corresponding to the periods 

 1786-95, 171)5-1805. and 1805-10, which were not observed, and are 

 therefore necessarily the results of calculation. 



Year of Perihelion Time of Revolution 



Fasug*. In darn. 



1788 



1789 1211-79 



1791 U1J-67 



1795 1312-55 



1799 1213-44 



1802 1212-33 



180J 1. 



1809 1211-10 



1813 1812-00 



1815 1111-89 



1819 1211-78 



1823 1311-611 



1835 1211-55 



1829 1211-44 



1833 1211-32 



1835 1211 .'J 



1838 1211-11 



1842 1210-98 



1845 1210-88 



1848 1210-77 



1852 1210-85 



1855 1210-45 



1858 1110-44 



These results indicate that the times of the successive revolutions are 

 gradually shortening at the rate of about ,\i.ths of a day, or somewhat 

 more than two hours and a half. No other comet has hitherto offered- 

 any evidence of a similar shortening of tho time of revolution. Tho 

 existence of a resisting medium cannot therefore be considered as 

 established beyond doubt 



It might be supposed that the effect of a resulting medium would bo 

 to lengthen the time of the comet's revolution, rather than to xh< 

 as observation indicates. It is true, indeed, that 1 1 

 such a resistance is to retard the orbit. il motion of thu comet, and to 

 far to prolong the time of revolution ; but on the other hand, this 

 diminution of the tangential motion allows the central body to act 

 with greater efficacy in drawing the oomet towards the centre! 

 according to the theory of central forces, the nearer a body approaches 

 the centre of force the less must be the time of a complete i< volution. 



The following ore the elements of this comet corresponding to tin- 

 perihelion passage of 1858 : 



Mesa longitude 157 59' 18" 



Aicemling nodo 

 Mean anomaly . . 

 Inclinttion . 

 Longitude of perihelion 

 Dean daily motion 



114 18' S4" 



J 1'4".0 

 13' 4' 15".0 

 157' 47' 80" 



1074".05 



Available for 1858, Oct 18'5, Berlin mean time and mean equinox. 



