No. 3.] MINOR PLANETS— LEUSCIINER, GLANCY, LEVY. 147 



coordinates are designated by e or / and r. The relations which are analogous to the above are 



■eo sin £ = c„ + n„t+ng8z = c„ + no{t + Sz) /l+c, 



f cos/=Oo (cosi — *!o) f sin/=fl„-y/l — ''o^ sin £ 



f=v—7t„ r — fO +v) 



These are the equations for motion in the orbit based on constant osculating elements and 

 appropriately determined constants of integration. 



If, in place of osculating elements and Hansen's ndz and v, v. Zcipel's elements and the 

 corresponding perturbations are used, the equations are the same in form. In v. Zeipel's 

 notation s and / take the place of I and /. The omission of the dash over these variables is 

 permissible, since the physically real values, with which they might be confused, do not occur 

 in the theory except for the date of osculation, where the subscript zero is added. It is to 

 be noted that, through v. Zeipel's choice of elements, the coordinates and the perturbations 

 have values which are numerically different from the Hansen cjuantities of the same designation. 



Let the time be the date of osculation and denote the true coordinates by £„, v^, r,. Then 

 the preceding equations for undisturbed motion become Z 121, equations (206), (207), and 

 Z 125, equation (230). 



Let the disturbed eccentric anomaly and radius vector (s, r) be £, and r„ respectively. 

 The relations for disturbed motion become Z 121, equation (209), and Z 122, equations (210). 



The first derivatives of these expressions are given by equations (208) and (211), respec- 

 tivelj', and the time rate of e is given by the equation following (209). 



The solution of the four equations (210), (211), with the aid of all the others, determines 

 the four unknown constant elements, a, e, n, c, or, better, a — a^, e — e„, tz — ;r„, and c. 



The fact that the adoption of the new elements in connection with the perturbations ndz 

 and V, as developed in the preceding sections, is equivalent to the use of osculating elements, 

 follows from the simultaneous solution of the equations for the disturbed coordinates and their 

 velocities and the corresponding equations for undisturbed motion. 



The method of calculating c from the equation 



. ,, , ,o (• = £, -e sin £i-W(?2 



IS given in the example, page 18. 



After many laborious transformations the other three unknowns are expressed in terms of 

 familiar functions in equations (233)-(23G). In the verification of these equations slight differ- 

 ences in the numerical coeflicients of certain unimportant terms were found. The magnitudes 

 of these coefficients depend upon the number of the terms included in making the transforma- 

 tions. Since it makes little difference whether or not they are included and since v. Zeipel's 

 values present a more symmetrical form of a later auxihary function, we adopted his coeffi- 

 cients. 



In the functions x, y, z the arguments and factors are functions of ij, r., £„ ^„ J, I, where 



but at the beginning of the computation only j;„, r„, f„, tf„, J,,, J^, the corresponding functions of 

 osculating elements are known.' 



' Thorr is a confusion of notation in v. Zeipel's developments. In Z 127, et|uation (238), flo is defined to be the value of at the date of oscu- 

 lation when osculating elements are used for the planet, and ei signifies the argument if the elements a, e, n, etc., are employed, or bv Z 9 

 equation (43), ' ' ' 



and their diTerencc is computed by Z 127, equation (238). 

 In the collection of formulae by Z 133, 



This U an approximation (or the above equation. 

 Again, mZ, 60, 



If the secular terms are counted Irom the date of osculation, the (actor (.o—et) ought to be replaced by (o—M. 



