MINOR PLANETS DISCOVERED BY WATSON LKrsCIINER. 205 



Then the nonsecular portion of the perturbations becomes for a particular numerical value 

 of i, i now positive only, 



ndz = a +li) sin ig + a- i(l sin ( — ig) + & +(t) cos ig + b- (i) cos (—ig) 

 = (a ni> -a. (i) ) sin ig + (b +lil + 6_ (t) ) cos ig 

 =a t sin ig + bi cos ig 

 where 



a t =I i -m +i i cos ( M +i t —i'g") —S^rnr't cos (M~\> — i'g') 

 b t = 2 i -m + \- sin ( M + \ -i'g') +I i >nr l i - sin (M^-i'g') 



if Bl be again omitted in v and w/cos i. 



The o, and 6, are therefore functions of the original A\ and B\> coefficients and of the argu- 

 ment g' . They are tabulated for each positive value of i at intervals of 6° of g' from g' = to 

 ^'=360°, or where the perturhations vary rapidly at intervals of 3°. The difference in the 

 a, and b, Tor one degree is also given, higher differences having been considered when necessary. 

 The constant B I occurring in v and u/cos i is included in b . It is not necessary to compute 

 a since this is multiplied by sin OXg. 



SECULAR PORTION OF THE PERTURBATIONS. 



This is of the form 



T{D„ + IfOi sin ig + I^D, cos ig} 

 where T=t—t in Julian years, and may be written 



Tc 



where 



c =D + IjCj sin ig + IfDt cos ig. 



c is a function of g and is tabulated for every 6° thereof in Table V. 



g was kept explicit in preference to g , as more terms could thus be combined in the tabu- 

 lated values of a, and b,. These coefficients were also found to converge more rapidly with g 

 explicit. 



The complete tabulation of the perturbations of this" group of eight planets is, therefore, 

 in the form 



.T t a, sin ig + -fit cos ig + cT 



for each of the components \\dz, log (1 +v) =3 log /■, and u/cos i. 



ARRANGEMENT OF THE TABLES. 



Table I gives the values of the mean anomaly g for January 0.0 of every common }*ear and 

 for January 1.0 of every leap year from the date of discovery to 1930, and their changes for the 

 different months and days. 



Tabh II gives the coefficients a t and /*, of the periodic parts of the perturbation ndz with 

 the argument g' in units of 0?001 and one decimal thereof. 



Table III gives the coefficients a t and &,• of the periodic parts of the perturbation log 

 (l+v)=d log r with the argument g in units of the fifth place and one decimal thereof. 



Table IV gives the coefficients a, and 6,- of the periodic parts of the perturbation u/cos i 

 with the argument g' in units of 0?001 and one decimal thereof. 



Table V gives the coefficients c of the secular parts of the perturbations for all three com- 

 ponents with the argument g. 



Table VI gives the constants for the equator for the beginning of every year from the date 

 of discovery to 1930, inclusively of the logarithms of the epiantities cos a, cos b, and cos c, by 

 which the perturbation 3p must be multiplied to obtain the corrections Ax, Ay, and Az to the 

 heliocentric ecpiatorial coordinates x, y, and z. 



