686 CAVALLIN, SECÜLAR PERTÜRBATIONS. 



The condition (P) is valid, and therefore also the formula 

 (I"), for all planets within our system, with the exception of the 

 Earth, Venus and ahout a dozen of the sinall planets. 



Again, Avhen the condition (!') is not satistied, no explicite 

 formula is up to this time given for the determination of the 

 mean motion. 



Generally it is said, that no mean motion exists in this case. 



In the foUowing it will however be demonstrated, that at 

 all events there exists a mean motion, limited by the greatest 

 and the least gi . Methods will also be given for its deter- 

 mination. 



The Solution of the above system of equations (I) is a 

 special case of the Solution of the following purely mathemati- 

 cally formulated question: 



Whe7i 



.pßi. 



2 ^.6'^'-' , (1) 



where the constants A,. are all signiftcative and in general ima- 

 ginary, the constants g all real and t a real variable; to deter- 

 mine the modulus r and the argument ß as functions of t. 



No loss of generality arises, if we assume the constants g ,. 

 all unequal, for if two or more terms contain the same g^ these 

 terms may be joined to one. 



Besides we at first assume all the g positive and put 



9\> 92---> 9n- (^) 



As convergents to the constants 



9l^ 92^ ■■•^ Un (3) 



we assume the fractions with common denominator 



7)1 ' m ' ' ' " ' 7W ' 

 which we suppose to be reduced to their lowest terms. 



