121' Sir J. W. Lubbock on some Methods of developing 



R t>1 = B + 2? 2 cos 2 i + B A cos 4 i + &c. 



R^i contains only B , U 2 , B 4 , &c, having even indices. 



R tj2 = J3 cos i + B 3 sin 3 i + B 5 sin 5 i + &c. 



R, j2 contains only B v B 3 , B 5 , &c., having uneven indices. 



R J}3 = ^4 2 sin 2 i + -<4 4 sin 4 i + -4 6 sin 6 « + &c. 



R i3 contains only A 2 , A 4 , A 6 , &c, having even indices. 



H i4 =A x sin i + ^ 3 sin3 i + A s sin 5 » + &c. 



R t - 4 contains only ^4 1} ^ p , A 6 , &c, having uneven indices. 



I have thus endeavoured to furnish expressions for obtaining the 

 coefficients in the development of any series of sines and cosines by 

 eliminating successively all the coefficients of sufficient magnitude 

 to be sensible except the one sought. I feel strongly impressed 

 with the truth of the opinion* of Le Verrier, and I think that, if 

 not altogether impracticable, it is at least exceedingly difficult in 

 the case of the smaller planets and comets to obtain the coeffi- 

 cients in the functions which occur in the theory of the perturbations 

 by any method which is not founded upon particular values of the 

 function, that is either by the methods here treated, or by the 

 method of mechanical quadratures. If the former are employed, 

 the tables of logarithms of particular values of certain quantities 

 which I have described elsewhere, due to particular values of the 

 eccentric anomalies of the planets, would be useful, and would reduce 

 somewhat the labour of calculation of the perturbations, the par- 

 ticular values of the eccentric anomaly being multiples of the 

 arbitrary angle, which I have for simplicity made equal to unity ; 

 or if the method of quadratures is to be used, the particular values 

 of the eccentric anomaly being multiples of 10°, or other part of 



the circumference — , n being a whole number. 



* L'interpolation parait done seule susceptible defournir les coefficients correspondanis 

 a des multiples eleves des longitudes moyennes. Les calculs sont encore assurement tres 

 longs ; mais ils ne sont pas impracticables comme ceux qui resultent des developpements 

 algebraiques. — Le Verrier, De'veloppements, 1, p. 6. 



