202 MEMOIRS NATIONAL ACADEMY OF SCIENCES, VOL. X, NO. 7. 



tables of each planet, by a { sin ig and b t cos ig or a t sin k and b { cos ie. They are to be entered 

 with a=a i and A = ig or is as arguments. The products are tabulated for every degree of A 

 from 1 ° to 90°, and for every unit of a from 1 to 100. For a = 1, the products are given to five 

 decimals: from a = 2 to a = 9 to four decimals, and from a = 1 1 to a = 100 to one decimal. This 

 latter part of the Traverse Tables has been copied from Table II of The American Navigator 

 by Nathaniel Bowditch. The a t and b { coefficients are tabulated to one decimal of the 

 adopted unit of 0?001 for noz and u/cos i, and to one decimal of 0.00001 for dlog r. The 

 products may be taken directly from the tables by double interpolation for values a t and b t up 

 to 100.0 units in each case, i. e., for coefficients not exceeding 0?1 or 0.001 , respectively. If the 

 coefficients be larger, the products may be conveniently found in parts with the aid of that 

 portion of the table which is given to four and five decimals. Example : Required the product— 



a sin A = - 2479.6 X sin 306?3S ; Unit of a = 0?001 



Since the angle is in the fourth quadrant, the algebraical sign of the product is +. The 

 numerical part may be taken from the Traverse Tables in the form : 



(1000 X 2 + 100 X 4 + 79.6) cos 36?3S) 



1000X2 cos36?38 =1610.1 Table B, page '-'is 



100X4 cos 36.38 = 322.0 Table B, page 219 



79. 6 cos 36 . 38 = 64.1 Table B, page 230 



-2479. 6 sin 306. 38 =1996.2 units=l?9962 



With Bremiker's five-place tables the product is found to be 1996.3, which agrees with the 

 former result within the accuracy of the computation. 



ELEMENTS (TABLE C). 



In this table are given the final elements of the twelve planets and the quantities m and g 

 for computing the magnitude at opposition, the latter being taken from the Berliner Astro- 

 nomisches Jahrbuch. 



SPECIAL TABLES FOR THE TWELVE PLANETS. 



The special tables for the twelve planets are arranged in three groups. 



The first group contains eight planets for which the argument of the developments is 

 (ig-i'g'), and for which the perturbations are tabulated in the form: 



2',(«j sin ig + b t cos ig) +cT 



where the coefficients a u b ( , and e are functions of coefficients of the original developments; 

 a { and b t are also functions of g' , while e is a function of g. 



The second group contains three planets for which the argument of the developments is 

 also (ig-i'g'), but in which all terms having (ig-i'g') or a multiple of (ig-i'g') as argument 

 are combined in the tables for particular values of i and i' under a single argument (ig-i'g'). 



If we denote the three components ndz, log (1 + v) and <?/?, by the symbol j, then the per- 

 turbations are tabulated in the form 



r = -iri + rtT-c'r 



where i is the numerical designation of the various arguments and c r is a constant. 



The third group contains but one planet, (93) Minerva, for which the argument of the 

 developments is (i—i'/i)e—i'(g' —fig), ami for which the perturbations arc tabulated in the form 



2'i<ii sin ie + J,-&,- cos is + (Jl>„ ), 



where the a t and b t are functions of coefficients of the original developments and of the argu- 

 ment N = s-g'-/ie sin s, and where (J6„) ( is the nontrigonometrical secular part of the per- 

 turbations. Explicit directions for the use of the tables and an example are given with each 

 group. 



