128 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv. 



The remaining terms in the differential equation for ndzy" are, by eq. (143), 



|^{n52 w- (7i52 W)} = (1 _e COS £)[lf/(^) + n^p)-|(F/')-i =•/')¥ tfi(') + ^=-.('))j 



-[(1-e cos e)[F/(')+r^F)-|(^FW-i^/'))(F,'» + ^^,('))n 



aU the terms of which are of the second order whose lowest rank is the second. They therefore 



contain the factor — ^ • 

 vr 



To obtain ti^s/^' it is necessary to return to eqs. (124)-(130) and make developments for 



terms of the second order similar to those for first order. The resulting differential equation is: 



(1-e COS£)r_^_,„ ,„^. „n,Ail?( 



f'NQ ''1 ' ' ^V-*- t> WO C y #V M/ OXXJ.C B, o 



,ndz^(2) ^ ^' «^"^^' |^^2,(" - (n52,(')) }^, IF/" - (1 - e COS £)>) w sim.^. F/^' 



_ra__e^os_£) ^^^^^(,) _ (^^^^(,)) } 1 17^(1) -d-e cos £)j? m; sins^^ F/^'l 

 -^[(1 -e cos F.("]^{7i52.- (nfe/-))} -|^(n5z,«)- 



The sum of the last two equations, when integrated, gives the terms of second order having 



Factor — ~ 



ur 



Ivticallv that 



the factor — j- It has been shown by v. Zeipel through computation and we have shown ana- 



and 



Therefore, 



d 

 de 



F/(^'+^{n52/')-(naz,('>)}^F/" = )j w sins^^ F/^) 

 [{l-e cos e) W')]^ {n52i<"- (ri,fe/»))} +w^(7i(52i<=') =0- 



{w5z.(=) - (71^2/=') +n32,^'^} = (1 - c cos £) j[ F,](=" - 1/ F/') - 1-^/")( l^V" + -J.= /") 



-[(1-. cos ^)jn^F-|( W')_|^,o))(^F.(')+is/'))j] 



The integral is tabulated in Table XXXIV. 

 Summarizing, we have included first order terms in 



ndzi + ndz^ + ndz^ 



given by tables XXX, XXXI, XXXII and second order terms in 



ndZi + ndz^ 



given by Tables XXXIII and XXXIV. The addition of Tables XXX-XXXTV gives the short 

 period terms in ndz, or, the function 



ndz — \_nSz\ 

 which is tabulated in Table XXXV. 



Returning now to the differential equation for ■d, the evaluation of F (i?, s) and its derivatives 



in Z 78, eq. (127) gives Z 91, eq. (146). The variable does not appear; -j- is a function of 6 alone 



Therefore the function is of long period. The integration is one step in the determination of 

 \ndz\, the long period terms in the perturbations of tlie mean anomaly. 

 '(1 — e cos £) W\ is tabulated in Table XXIX6. 



The function 

 The function 



{\—e cos c 



is given in Table XXXVI. 



i)( IF- ^S'Y F+ \^S\ computed from Tables XXIXo. and XXIXc, 



