27G NEW FORMUIiiE RELATIVE TO COMETS. 



hope, to attention and preference. Their mode of investigation I also 

 judge important, and as peculiarly eligible. It has enabled me to ex- 

 hibit the formulae hitherto given, as particular states of those just no- 

 ticed ; and besides others equally simple, it has furnished two new and 

 general sets of expressions for the exceptive cases in which the observed 

 latitudes and longitudes of the comet would render the general formulae 

 doubtful or indeterminate. To the analysis of the principal of these 

 results, and Avith regard to practical applications, I have adjoined the 

 data of the comet of 1805, and for which I am indebted to the excel- 

 lent treatise of Mr Pontecoulant. The corresponding velocities I have 

 computed by the formulae now given, and by others connected with 

 the method of La Place. Their comparison has led me to some re- 

 marks with which I conclude this paper, and which I have inserted 

 from an opinion of their analytical and practical importance. 



Before I enter on the proposed investigation, I think it may not be 

 improper to observe, that within a few days, I have been favoured with 

 the perusal of Mr Encke's Astronomical, Annual Register (tdstrono- 

 mische Yahrbuch) for 1833, in which its distinguished author has 

 given a full and neat analysis of Dr Olbers' method of determining the 

 orbits of comets. The greater part of that analysis I had in fact the 

 earlier pleasure of reading in Dr Bowditch's Appendix to the Third 

 Volume of his Translation of the Mecanique Celeste. But I had not 

 been previously apprized of Mr Encke's remarks on methods which 

 differ from that of Dr Olbers ; and in this paper, I would be under- 

 stood as having no wish to aim at lessening the predilection with which 

 I am now acquainted, and which may be well and reasonably founded. 

 My own mathematical partialities, I am not unwilling to avow. I 

 entertain them on methods and processes of computation which furnish 

 symmetrical and direct results ; and in no slight degree am I favourable 

 to that method which is connected with the formulae now occupying 

 my attention, and which I have endeavoured to present in such form 

 as to merit the approval of Mr Encke, and the author of the Theorie 

 Analytique du Systeme du Monde. 



Consider a comet at any point C in its orbit, and let its place at the 

 distance r from the centre S of the sun, be determined by the rec- 



