MINOR PLANETS DISCOVERED BY WATSON— LEUSCHNER. 211 



The tables are based upon the "General Perturbations of Minerva by Jupiter," published by 

 Eichelberger in the Memoirs of the National Academy of Sciences, Volume III, Tbird Memoir. 



Except for some minor changes, due mainly to the adoption of a different epoch and of a 

 slightly different value of Jupiter s mass, the perturbations and elements given on pages 361. 

 362, and 363 are the same as those originally published by Eichelberger. 



The expressions ndz, v, and u/cos i are developed in the form: 



ndz = nz -g a - nt = ZfZ^A \> sin [ (i - i'ft)e - i' (g' - fig)] + 

 ZilfB): cos [(i-i',i)s-i'(g' -fig)] + 

 ■ (t-t ){IiCi sin ie + liDi cos ( s} 



v-Bi+ItlrAl ^n [{i-i' n)s-i'{g' -fig)] + 



Zili&t cos [(i-i'fi)s-i'(g'-fig)] + 

 (t - t a ) {D + Z 4 Ci sin is + ZiDi cos is } 



u cos i=Bl + Z i Z?A\> sin [(i-i' n)s-i'{g' —fig)] + 

 liZiB 1 , cos [(i-i'fi)s-i'(g' -fig)] + 

 (t - t ) {D + liCt sin is + J,Z>,- cos is } 



i varying from — oo to + <x> , and i' from to oo , except that the constants Bl and D n are 

 segregated from the double sums. 



Before tabulation, the expression for v was transformed into an expression for log (1 +v) = 

 3 log /• by multiplying the expression for v by Mod. sin 1", the higher powers of v being inappre- 

 ciable within the accuracy of the tables. Similarly the expression for u/cos i was transformed 

 into an expression for 3 t 3 by multiplying the expression for u/cos i by a sin 1". 



For tabulation the perturbations are segregated into their secular and nonsecular portions. 

 The long period term in ndz is 



+ 11". 1 cos [5s - U(g' + ue sin s)] + 4".() sin [5s - 1 3 (g' + fie sin s)] 



and is also segregated from the periodic portion. 



NONSECULAR PORTION OF THE PERTURBATIONS. 



The nonsecular portion of the perturbations has the form 



Bl + ZiZfM. sin [ti-i'u)s-i'(g' -pgfi+SJfB, cos [(i-i' fi)s -i'(g' -g)] 



for all three components. For ndz, Bt and Z) IJ =0, these constants being contained in the 

 elements g and n, respectively. 

 By writing the argument 



(i — i'fi)s — i' ig' — fig) 

 in the form 



i'(s-g' -fie sin e) + (i-i')s = i'A T + (i-i')e 

 where 



V= (g -g') + (s -g - ae sin s) = (g -g') +J(g -g') 

 and where 



J (g — g ) = s — g — fie sin s 



we obtain for the nonsecular portion of the perturbations the expression 



SiSgA^ sin [i X + (i - i' )s] + ZilfB'e cos [i' N+(i- %' )e] 



where B\ is omitted for the present. 



If we write (i) for (i— %'), and expand the sines and cosines, the expression becomes for a 

 particular value of (i) 



Z i 'A" i '. +i ' sin i'N cos (i)s + Z V A^.* V cos i X sin (i)s 

 + Zi-BW,+* C os i' N cos (i)s - J' i 5';. ,+ '" sin i' N sin (i)s 



= cos (i)eZi{Ay +i ' sin V X + B'^*' cos i'AT + sin (i)eZ i '[A^ +e cos i'N-B",^'' sin i'N] 

 where, as before, i' is always positive, and where (i) = (i—i') may be either positive or negative. 



