THE ORBIT OF NEPTUNE. 



CHAPTER II. 



PROVISIONAL THEORY OF NEPTUNE. 



§ 6. All the perturbations have been computed by formulos founded on the 

 method of La Grange ; the development of the perturbative function in series, 

 and the variation of arbitrary constants. 



The following notation is used : 



I =: mean longitude. 



/I =z mean longitude, counted from ascending node of inner planet on outer one. 



^ ir: inclination of orbit to the ecliptic. 



y =z mutual inclination of two orbits to each other. 



a r= ratio of the mean distances. 



u =r sin i y. 



f =: mean anomaly. 



6) := distance of the perihelion from the ascending node of the inner planet on 

 the outer one. 



For the other elements the almost universal notation of astronomers is adopted. 

 The elements which pertain to the outer planet (Neptune) are distinguished by 

 an accent. 



The potential of the disturbing force exerted by one planet upon another, usually 

 called the perturbative function, may be developed into an infinite series of terms, 

 each of which shall be of the form 



m - cos {i'2/ + ■i^ +/"' +/") 



in which i, ^', j, and / are numerical coefficients. 7i is a function of the ratio of the 

 mean distances, the eccentricities, and the mutual inclination of the orbits. 



Then, by the theory of the variation of arbitrary constants, any term of the 

 perturbative function in the action of an inner on an outer planet will cause the 

 following differential variations of the four elements which determine the form 

 of the orbit, and the position of the planet in it. Putting 



■i'/l' + ^'/l -{-j'a' -\-Jc)z= N, 

 e = sin ip, g z^ cos 4 tan 1 4' 5 



we have -^ izi — 2 mi'ha'n' sin N. 



at 



f^e' 7 f • , ■ ) • ,r 



-J- =z mn'li < j' cot n^' + ^y r sni A, 



-f- iz: mn cot iL' --— cos A. * 



at ^ dd 



