No. 3.] MINOR PLANETS— LEUSCHNER, GLiVNCY, LEVY. 89 



The f miction <^i is given by Z 59, eq. (103), or, 



From the table of T",, page 82, it is not difficult to Avrite immediately 



\[X,]-r,[Y,])dO 



3 

 T 



r 



The terms of higher order must be obtaincil by the usual method for the mechanical multi- 

 plication of series. A logarithmic multiplication is the most direct. 



In each term in the expression for 0i the terms of lowest rank must be of the first rank. 



Recalling the tabulation of factors in Z 53, w, — . — j, — ^, etc., are all of first rank. But the 



coefficient for a given argument consists of three terms in ascending powers of w. Hence 

 (^, — w, within the limits of the given tabulation for T^, is of rank 1, 2, 3 for each order in the 

 mass. Table X\T:, giving cpi — w, is tabulated with double headings. The three subheadings 

 indicate the expansion of the coefficients in a Taj^lor's series and the main headings give the 

 factors in the development of the radical in Z 59, eq. (103). 



Having found <^i, its reciprocal, 4>i~\ inclusive of first order in the mass, is given by 



'l>r-' = i-^f ([X,]-ri[Y,])dO 



The second term is the negative of the first three columns of Table XYI multiplied by itr^. 



The product of 2<l>~'^ and that part of T^ which contains cp gives -^, and integration with respect 



to 6 gives «!, tabulated in XVIII. The function Uj is of first and higher rank because the factor 

 <^,~' is of rank minus one and T^ is of second rank. 



From Table XM^II j/i can be read by inspection, and j^i/j added to Table X\T gives 2,, 

 tabulated in Table X\ai. The function IF, is the sum of Tables XVII and XVIII. 



In the integration those terms whose arguments are independent of d are of the nature of 

 constants. In accordance with the condition that there may be secular terms in 6, the integral 

 contains such terms as the following: 



O-Jc sin ((P + J). 

 As the constant of integration 



do-ksin {(f' + J) 



is added. Hence the integral contains terms such as 



(O-Oo) t sin ((l> + J, 



where ^0 is the value of for the time t = 0. 



In passing, it should be noted that, in order that the expansion of Z 59, eq. (103), shall 

 represent the function, wo must have 





{[X,]-r,[Y,])dO 



< 1 



and this condition should be tested for a given planet before applying this method of determining 

 the perturbations. 



To the computer the extent of auxiliary tables, the arrangement of series in logarithms or 

 natural numbers, in seconds of arc or radians, inclusive or exclusive of numerical factors, and 

 foresight in combining operations — all these are of the greatest importance. But considerations 

 of this kind would carry the reader into complicated details which are best left to the com- 

 puter's own judgment. 



On the other hand, general considerations about the extent of the pubhshed tables are of 

 importance in the discussion of the accuracy of the final tables. Yet, for a given limit of 

 accuracy, it is so difficult to determine, for each tabic, the highest powers of vi', w, t), r/, and f 

 that little or nothing is said about it in connection with individual tables, but the discussion 

 is reserved imtil later. 



