Bowditclvs Remarks on Doctor Stewart's formula. 



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supposing with the author the exceiitricity to be very small, or 

 the orbit nearly circular. 



There is now no great difficulty in settling this point by means 



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of the analytical expression of the motion of the apsides given by 



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La Place in his excellent Lunar Theory, publisKed in the third 



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volume of his Mecaniqiie CelestCf where however the approxima- 



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tion is carried on to a higher order of the powers and products of 

 the excentricities and disturbing forces^ than is absolutely neces- 



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sary for the present purpose. Upon applying this method^ I 

 found that Doctor Stewart's formula was very far from being so 

 correct as had been supposed by most of the writers above men- 

 tioned. On the contrary^ it appears to be essentially defective. 

 The first and most important term of the series is double its tru5 

 value; and the whole formula gives an accurate numerical result 

 only when the motions of the primary planet and satellite have a 

 certain proportion to each other, which (by a remarkable coinci- 



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dence) happens to be the case (nearly} with the moon and earth; 



but the same formal 



ould not answer if the 



I's distance 

 it now is ; 



from the earth was much greater, or much less, than 



and it would require but a very small decrease of the moon's mean 



distance from the earth, to render the sun's distance infinite when 



puted accordin 



Doctor Stewart's directiu 



so that this 



method would have failed, if it had been applied to other cases, as 

 for instance Jupiter's Satellites.* To prove this I shall now 

 proceed to reduce the formula given by La Place to the case com- 

 puted by Doctor Stewart.* 



The orbits being supposed nearly circular and situated in the 



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* Neglecting the mutual action of the satellites, and the effect of the oblafe- 

 ntss of the primary planet- 



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