﻿Prof. P. Lowell on the Asteroids. 



349 



three fulfilling the criterion in the gap n = 2n f . This gap 

 then, also, cannot be explained by libratory action. 



15. We shali now proceed to show that the criterion, though 



theoretically correct so far as the terms we have considered 



go, does not hold throughout the action. To see this we 



may notice that theoretically the libration is possible to any 



angle short of 180° from the mid-point dependent only upon 



dB 

 the values at the epoch of osculation of -j- and A. And 



calculation shows that this angle decidedly exceeds 90° in 

 the case of (S), U<>«), an d (132), that angle being /3 where 



00 2 =iA[cos(0- 7 )-cos/3j. 



But if — ry = at j3, the right side of the criterion vanishes, 

 so that at some point the criterion may state that there is 

 libration, whereas at an earner stage of the action it stated 

 the contrary. This means that a libration in the case is not 

 a true libration, but seeming such within limits only, to be 

 replaced later on by continuous movement of (# — 7). 



16. That this is actually exemplified by the asteroids we 

 shall perceive from ihe following table of the elements of 

 (122) for various epochs since its discovery. 









Table V. 









(S) 



0. 





■w. 



<p. 



P- 





1872 Jan. O'O 



—97° 



26' 



208° 



12' 



2° 



5"3 



613-52 



Peters 



1875 Jan. 1-5 



-82 



25 



208 



38 



2 



7'8 



614-19 



Stockwell 



1876 Apr. 5. 



-80 



3 



208 



53 



2 



8-9 



614-12 



>) 



1878 Sept. 2. 



-79 



11 



205 



49 







15-8 



61474 





1879 Nov. 16. 



-78 



4 



204 



53 



2 



19 1 



61572 



» 



1830 Dec. 20. 



-76 



27 



204 



28 



2 



19 7 



615-57 





1883 July 12. 



—72 



58 



203 



46 



2 



22-5 



614-74 





1884 Sept. 24. 



-74 



46 



200 



40 



2 



28-5 



614-40 



Oppenheim 



1885 Dec 8. 



-72 



39 



200 



52 



2 



30 



614-82 



Tietzen 



1887 Feb. 21. 



-70 



5 



201 



11 



2 



30-9 



615-03 



1? 



1888 May 6. 



-68 



36 



200 



45 



2 



35 



613-90 





1889 July 20. 



-68 



51 



198 



49 



2 



40-9 



61424 



Lange 



1890 Oct. 3. 



-68 



26 



197 



13 



2 



44 



615-66 



M 



1891 May 15. 



-63 



34 



195 



54 



2 



46 7 



61472 





1895 July 29. 



-61 



26 



195 



53 



2 



47'7 



614 77 





1896 Oct. 11. 



-59 



14 



196 



10 



2 



46-1 



615-15 



M 



1897 Dec. 25. 



-53 



14 



196 



37 



2 



45\S 



61552 



?J 



1899 Mar. 10. 



-55 



21 



195 



54 



2 



52 -4 



61511 



SJ 



1901 Aug. 7. 



-54 



59 



192 



34 



2 



59-9 



615-80 





1904 Jan. 4. 



-51 



54 



191 



30 



3 



4 



615*60 



M 



1905 Apr. 28. 



-49 



48 



191 



7 



3 



1 -5 



615-07 



n 



1907 Sept. 25. 



-45 



50 



189 



41 



3 



3 -8 



61537 



,, 



1908 Oct. 29. 



-44 



19 



190 



46 



3 



5-4 



615-77 



. 



1911 May 7. 



-40 



56 



189 



54 



3 



11 2 



61437 



» 



