No. 3.1 MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 81 



Our problem is now the integration of the partial diiTerential equations Z 7, eq. (33), Z 8, 

 eqs. (37) and (39), and Z 9, eq. (47'). 



In the trigonometric series to be integrated the argument is a function of d, e, (p, J, I. 

 The last two are constants. According to the principles of Hansen, ([/ occurs outside the opera- 

 tion. Numerically, however, it is ec^ual to e. The argument 6 contains s implicitly. See Z 9, 

 eq. (43). Hence we must, in general, write 



F(b, 6) 

 and 



di ds dd ds 



In order to set up the partial differential equations from the total derivative, the following 

 notation is introduced: 



F(£, d) = [F{e, 6)] + F{e, 6) - [F{t, 0)] 



where [F{s, d)] signifies that part of the function which is independent of s. Again, since b has 

 the period of the planet, there can be no secular terms in s (with the exception of the function d), 

 i. e.. 



On the other hand, the argument d varies much more slowly, and there may be secular terms 

 in 0. Hence 



D 



m 



^ Mo 



and may occur outside the sign of integration. 



Owing to the presence of the required function in the differential equation, the integrations 

 must be performed rank by rank where rank is defined as follows: 



In the course of the developments there arise negative powers of w. Since w is a small 

 quantity, these factors increase the numerical value of the terms, or, in other words, they lower 

 the order. Therefore, it is better to define order in terms of both the disturbing mass m' and w. 

 For this purpose v. Zeipel makes the assumption that both w and ■^jm' are quantities of the 

 first order. Order so defined is called " rank," and the word "order" is reserved as usual for 



7/1 '" 



the powers of m'. The factors — j- are arranged according to rank in Z 53. 



Any function is then written in the form 



F{,,d) = F,{,,d) + F,{e,d) + F,{,,e)+ 



where the subscript denotes the term of lowest rank, for F,- (=, 6) contains terms of more than 

 one rank since each coefficient is itself a Taylor's series in w. In assigning rank it is to be noted 

 that the coefficients in all the preceding tables contain the factor m' implicitly. The implicit 

 mass factor is indicated at the foot of each table which follows. 



On the basis of the foregoing principles, the differential equation for W, 



dWbW bW dd^ 

 de bs bd de 



expressed in Z 52, eq. (91), is broken up into four equations, Z 53, eqs. (95, — QS^), according 

 to rank, and before integration they are again subdivided according to parts which contain s 

 and parts which are independent of s. The total derivative is then in the form of eight equivalent 

 equations, and the integration can be performed in the following order: 



W,; W,-[W,]; [FJ; W,-[W,]; etc. 



It is possible to avoid the computation of T^, as v. Zeipel did, by the introduction of some 

 auxihary functions, but we found it preferable to tabulate them. 



Employing Table XVa, and by inspection of Tables "VIII, IX, X, XI, T, is written directly. 

 ( T has no terms of first rank.) 

 110379^—22 6 



