8 CELESTIAL MECHANICS: LEVSCHNER 



sion to encourage researches similar to Leveau's, and by supplying 

 samples of research surveys for a limited number of planets to pave 

 the way for a comprehensive international program in this connection. 



It cannot be too strongly emphasized that accurate osculating 

 elements are absolutely essential for fundamental investigations of 

 the perturbations. While this requirement is fully recognized, the 

 prevailing practice of changing elements for immediate ephemeris 

 purposes is apt to lead to erroneous interpretation of available 

 elements. Mean elements, in general, can be determined only after 

 osculating elements and perturbations shall have become available. 

 Some investigators have adopted as approximate mean elements the 

 average of elements published for more or less extensive series of 

 oppositions, assuming that these elements represent fairly reliable 

 osculating elements. Even if this were the case, it hardly ever occurs 

 that a sufficiently large number of elements, uniformly distributed 

 over the orbit, are available to guarantee that, in taking the average, 

 the effect of the periodic terms is entirely eliminated. But, as 

 previously stated, many of the apparently reliable sets of elements 

 are not osculating, but inferior elements produced by arbitrary 

 changes or with incomplete perturbations. 



Practically the only reliable method of arriving at accurate initial 

 osculating elements consists in representing the observations of a 

 limited number of oppositions by taking into account the special 

 perturbations and in testing the validity of the resulting elements for 

 one or more oppositions following. Osculating elements thus obtained 

 will rarely require later changes which would affect the coefficients 

 of the general perturbations. No correction of such elements should 

 be attempted, except on the basis of the determination of complete 

 special or general perturbations. As it was not considered necessary, 

 at the time, to adhere strictly to the foregoing principle in 

 Leuschner's program for the determination of the perturbations of 

 Watson's asteroids, allowances for slight inaccuracies may later 

 become necessary for some of the Watson planets. In particular, 

 corrections to the larger coefficients of the perturbations may be 

 necessary for the planets for which the initial adopted elements, con- 

 sidered at the time as sufficiently accurate, were neither accurate 

 mean elements nor accurate osculating elements. 



Attention has recently been called, in the Proceedings of the 

 National Academy of Sciences of 1921, Vol. 8, No. 7, p. 170, and in 

 the report of the Committee on Celestial Mechanics of the National 

 Research Council, Bulletin of the National Research Council, 

 Vol. 3, Part 4, No. 19, June, 1922, to the extremely satis- 



