204 MEMOIRS NATIONAL ACADEMY OF SCIENCES, VOL. X. NO. 7. 



Thus, since Jg„ appears in Table I and since the coefficients of the perturbations depend on 

 ^ , the correction An may be allowed for in the main by adding the same to the values of g as 

 given in Table I when the g are to serve as arguments. In cases where Jr is small and the 

 perturbations do not vary rapidly with g, the correction may be neglected. But with large 

 values of Jr. and rapid variation of the perturbations the omission of this correction may intro- 

 duce comparatively large errors of the second order in the residuals, through the consequent 

 inaccuracy of the arguments of the perturbations. 



TABLES OF EIGHT PLANETS.- -| 105 1 ARTEMIS. 1 115 1 THYRA, (128) NEMESIS, ( 133 1 CYRENE, (139) 

 JUEWA, <i6i> ATHOR. (174) PHAEDRA, (179) KLYTAEMNESTRA. 



The elements of these planets are mean elements. 

 .The expressions for ndz, v, and w/cos i are developed in the following form: 



ndz = nz -g - nt = I^i-A \< sin ( ig - i'g' ) + Z t 1^ cos (ig - i'g' ) 

 + (t-t ) { lid sin ig + Z \D t cos ig) 

 v=B° + 2 t Z*A!< sin {ig - i'g' ) + !£<&< cos (ig - i'g' ) 

 + (t-t ){D + lid sin ig + J 2 A cos ig} 

 w/cos i=Bl + lil^A 1 ,' sin (ig — i'g') + Z^Z^B',' cos (ig —i'g') 

 + (t-t ){D + I t a sin ig + I i D i cos ig) 



i varying from — so to +00, and i' from to 00, except that the constants Bl and D are segre- 

 gated from the sums. Before tabulation, the expression for v was transformed into an expression 

 for log (1 +v) =<5 log r by multiplying the expression for v by Mod. sin 1" and adding the higher 

 powers of v where they were appreciable. 



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

 portions. 



NONSECULAR PORTION OF THE PERTURBATION. 



The nonsecular portion of the expression is: 



Bl + IilfA 1 ^ sin (ig-i'g^+JjIfBi- cos (ig-i'g'). 



For ndz, Bt = D =0, these constants being contained in the elements g and n, respectively. Let 



A','=m! f cos Ml' 

 B\i =m'e sin M\> 



where m'> may always be taken positive. Then the perturbations may be written in the form 



Bl + ZiZf M'e m' e sin (ig-i'g' + MV) 



or also 



Z ( Z r m'f sin ig cos ( .!/;< - i'g') + ZfZ? m'< cos ig sin (Mi-— i'g') 



if B„ be omitted for the present. 



As a first step in the construction of the tables each coefficient m'- and angle Mi were com- 

 puted from the corresponding coefficients A\> and B\- of sin (ig — i'g') and cos (ig — i'g'). 



For a particular value of i the foregoing double sum becomes 



sin ig Z^ml^ cos ( M\- — i'g' ) + cos ig Z v m\> sin ( M \> — i'g' ) 



where i is to be taken both positive and negative, while i' is always positive. 

 Let 



a +0) represent the coefficient of sin ig. 

 a_,,, represent the coefficient of sin ( — ig). 

 6 +(() represent the coefficient of cos ig. 

 &_,„ represent the coefficient of cos ( — ig). 



