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time, are directly determined by the analysis. The solution to the 

 first power of the disturbing force is effected by means of the follow- 

 ing three equations, in which the letters have the significations 

 usually given to them in the planetary theory : 



dd h 



After substituting in the right-hand side of the first equation, the 

 values of r and given by a first approximation in which the disturb- 

 ing force is neglected, that side becomes a known function of t. The 

 equation can then be integrated approximately so as to give the de- 

 velopment of r in terms of t to the first power of the disturbing 

 force, and to any power of the eccentricity it may be thought proper 

 to retain. By substituting in that term of the second equation 

 which does not contain the disturbing force the value of r thus ob- 

 tained, the integration of the equation gives the development of 6 in 

 terms of t, and lastly by substitution in the third equation z is simi- 

 larly developed. The author has shown the practicability of this 

 method by obtaining values of r and to terms of the order of the 

 eccentricity multiplied by the disturbing force. The development of 

 the latitude, and a more particular application of the method to the 

 motion of the moon, are reserved for future consideration. The 

 particular advantages of this mode of solution are, that being free 

 from all assumption as to the forms of the developments, it gives 

 those which are alone appropriate to the problem, and it evolves 

 both the periodic and the secular inequalities by the same process. 

 Terms containing ent as a factor, which are met with in other solu- 

 tions of the same problem, do not occur in this method ; but there 

 are terms containing the factor e'nt, which are shown to be convert- 

 ible into periodic functions, and to have reference to secular varia- 

 tions of the eccentricity and of the motion of the apse. The paper 

 concludes with some general remarks on the principle of this approxi- 

 mate solution of the problem of three bodies, and an explanation of 

 the analytical circumstances which make it, in common with the 



