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



Let, in the foregoing expression, 



a-Hj, represent the coefficient of sin ( + (i)s). 

 (i- Ut represent the coefficient of sin ( — (i)e). 

 b +(i) represent the coefficient of cos ( + (i)s). 

 b- fi) represent the coefficient of cos ( — (i)s). 



Then the nonsecular portion of the perturbations becomes for a particular value of (i) 



(o +rt) — o_ w ) sin (i)£ + (b+ d) +b- (i) ) cos (i)e 



where (i) is always positive. 



Changing the notation by letting 



a { =(a +ii) -a_ (i) ) ; ft, = (b +li) -J_ (i) ) 



we obtain the following form for the nonsecular portion of the perturbations for a particulai 



value of (i), 



a, sin is + b t cos is 



where 



a, = 2VU +, ? +; ' cos i' V-5 + ';: + '' sin i'JV] - Z^A'^' cos i' N-B~^ +i ' sin i'A] 

 & < = 2 , ^ +( 4 9 + *' sin i'N+B + y +i ' cos i'A] + J,-[^-';. ,+ ''' sin i'N+B-^' cos i'iV] 



Tlie nonsecular portion of the perturbations have thus been reduced to the form 



Jjfflj- sin ie + Ifii cos is, i always positive. 



The a, and b t coefficients are functions of the original A\ and J5;< coefficients, as given on 

 pages 362, 363, and of the argument N, where 



N=g-g'+J(g-g') 

 and 



J(ij —g 1 ) =e—g — fte sin e. 



The a, and 6, coefficients are tabulated for the nonsecular parts of the perturbations with 

 the argument N in Tables II, III, and IV for all three components, the intervals being so chosen 

 as to facilitate interpolation. The difference for one degree is also given wherever necessary. 

 The constant Z?J occurring in v and w/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„ + 1 iC, sin ie + Ifi, cos is} 



where T=t— 1 , and may be written 



(A\) t + ZiiAaJt sin ie + Z t {dhi)t cos is 

 where 



UK), = TD n ; (Ja,) t = TO s : (Jb,), = TD,. 



The secular terms are thus reduced to the same form as the nonsecular terms. The values of 

 (Jb ) t , (Ja,),, and (J&,-)t are tabulated in Table V for all three components for the beginning of 

 every second year from 1865-1930, together with the changes for various intervals, to facilitate 

 their computation for any date within the table. The term (Jb ) t in ndz is contained in the 

 element n, and need not be tabulated. This heading has, therefore, been adopted for the long- 

 period term. 



THE LONG-PERIOD TERM IN rniz. 



The long-periodic term in ndz has been computed for every two years from 1865-1930, 

 and is tabulated in Table V in the block ndz under the heading (Jb ) t . Owing to its slow varia- 

 tion its changes are not given. 



The complete tabulation of the perturbations of Minerva is, therefore, in the form 



(Jb„) t +-!>, + (Ja,) t ] sin u + I$b,+ (Al,)d cos ie 

 for each of the components ndz, log (1 +v)=d log r, and dp. 



