Olbers on Comets. 161 



eighth degree. Dusejour has attempted to reduce every thing 

 to equations of the second degree, with what success we shall 

 see hereafter. Finally, Laplace has found means to apply a 

 mode of interpolation to several remote observations, so as to 

 obtain from the first and second differences the intermediate 

 places, at any required intervals, however small. His solution 

 depends on equations of the sixth or still higher degrees, and it 

 would perhaps leave little further to be desired, if the prepara- 

 tions, and the manner of interpolation, did not commonly re- 

 quire much more time, and labour, and computation, than the 

 solution itself. For further information respecting these me- 

 thods, the reader may consult Lambert, Insigniores Orbium Co- 

 metarum Proprietates ; Scherfer, Institutiones Astronomies The- 

 oretics; Lambert, Astr. Jahrb. Berl. 1777; Mem. Ac. Berl. 

 1771; Lagrange, Man. Ac. Berl. 1778, 1783; Astr. Jahrb. 

 Berl. 1783 ; Dusejour, M. Ac. Par. 1779 ; Laplace, M. Ac. Par. 

 1780. 



S 16. 



A general idea of the most useful of these solutions may be 

 obtained without any great difficulty. Considering the inter- 

 vals as infinitely small, we naturally assume, with Boscovich, 

 that the portion of the orbit concerned is a straight line, de- 

 scribed with uniform velocity. Hence the values of ^' and f may 

 be determined by a linear equation from 5" ; or §' =: Hj", and 

 ^'"r: Gg", H and G being known coefficients : and we may conse- 

 quently obtain the value of k' in terms of §'. The compa- 

 rison of the time with the space described then gives us the 

 expression k" »J r" = m T : and if we exterminate all the 

 irrational quantities, we come at last to an equation of the 

 form £A" * r" 2 —n'^l *], which is of the sixth degree, and is 

 the simplest that can possibly express the conditions of the 

 problem. 



i 17. 



Much as there is to be admired in some of these investigations, 

 and difficult as it may be to decide on their comparative value, 

 it will still be readily granted, first, that they are all more or 

 less imperfect approximations, requiring further correction, 



Vol. IX. M 



