No. 3.] MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 135 



to a numerical error or to the number of terms included. The remaining discrepancy is due 

 to slight inaccuracies of v. Zeipcl's computation. 



Table XXXI 11. — The discrepancy in this tabh> foUows from one in Table XVIIL Third 

 degree terms in Tabic XVIII were not integrated because, in the aggregate, they amount to 

 very little. 



Table XXXIV. — Our table is more extensive. Second degree terms are, however, not 

 complete, for they do not include second degree terms in 



[1J2] cos s + [22] sin e 



The discrepancies are of no importance. 



The integration of cq. (146) is best performed individually for each planet. The analytical 

 developments are as follows: 



The differential ecpiation can be WTitten 



^'^ /ft/oN^Y"' \ , '^/'"^ r >;} n\ 



By a change of variable 



Writing 



'^ ^..my '^"^ 





we have Z 96, eq. (152), in which the last term can be neglected. 



For a given planet the factors w, 1/, f and the argument i are known constants. There- 

 fore 1 +<^ («?) can be expressed as in eq. (153), as a Fourier series of sines and cosines of mul- 

 tiples of 2i>, in which the nontrigonometrical term is designated by <r. 



Expressing eq. (153 ) in terms of exponentials and solving for di-^e — {11 dz'] j by the expansion 



of {1 +<P («?)}"', and reintroducing the trigonometric functions, we have the equation 

 following eq. (153), in which the nontrigonometrical part is taken outside the brackets as a 

 common factor. The brackets in this equation do not have the special significance which 

 they have had previously. 



The variables e and i? are now separate and the integration can be performed. Trans- 

 ferring the common factor to the left-hand side of the eauation. nerforming the integration 

 and adding 



n , 

 — c — c 



n 



as the constant of integration, we have the argument 1^ expressed as a function of i? in eq. (154), 

 where ^ is defined by eq. (155). 



The reversion of the series gives ^ as a function of ^. We have by eq. (154) 



'^ = C + -^^cis"'^ 

 where ~0 is a small quantity. Given 



z = w + (r«l> (s) , where « is small, 



we have, by a theorem of Lagrange, 



F^^) = F(w)+a0(^v)F'(w) + ^^ ^[{<P{w)yF'iw)]+- ■ ■ ^^„[{(P (w)}''*'F' («;)]+ ■ • • 



By means of this theorem eqs. (156), (157) can be derived, where it is to be noticed that 



(^— gC + c') is an approximation for i!^ — !^^). In our developments we have used (C — Co)- 



