140 Royal Astronomical Society. 



missive power : that of sealing-wax, bees*-wax, sole-leather, glue, 

 gutta-percha, India-rubber, filbert-kernel, almond -kernel, boiled ham- 

 muscle, raw meat-muscle, appears to be unappreciable by the method 

 described. 



ROYAL ASTRONOMICAL SOCIETY. 



Nov. 12, 1852. — On a New Method of Computing the Perturba- 

 tions of Planets. By M. Encke. Translated into English and illus- 

 trated with Notes by G. B. Airy, Esq., Astronomer Royal. 



The main object of the author is to supply an easy and accurate 

 method of computing the perturbations of the various small planets 

 revolving between Mars and Jupiter, the number of which is con- 

 stantly increasing from fresh discoveries. The method, however, is 

 applicable to any of the other bodies of the planetary system. 



Let t denote the time from which it is proposed to compute the 

 action of the disturbing body. By a slight transformation the dif- 

 ferential equations of the motion of the disturbed planet are reduced 

 to the following form : — 





^°, y°. z° being the elliptical coordinates of the disturbed planet, 

 and r° its radius vector, as deducible from the elements of the orbit 

 in which it was revolving at the instant t ; X, Y, Z the resolved 

 values of the disturbing force in the directions of corresponding rect- 

 angular axes ; ^, rf, ^ the perturbations of x, y, z, counted from t ; 

 cr the perturbation of the radius vector; and k a constant quantity. 



These equations are integrated by mechanical quadratures, the 

 successive numerical values of the quantity under the sign of the 

 double integral in each equation being corrected by a peculiar process, 

 founded on the theory of finite dififerences. 



The first step is to divide the period for which the perturbations 

 are to be computed into a number of sufficiently small intervals of 

 time, each equal to t, and to calculate the values of X, Y, Z for the 

 times t~\ T,t-\-\ T,t-\-^ T,kc,y using for this purpose the elliptical 



coordinates x", y°, z° of the disturbed planet. As I, rj, 'C, — X 



AT 



y- are assumed severally equal to zero at the commencement of the 



period for which the perturbations are to he computed, the first 

 three values of the quantities under the sign of the double integral 



(or — . — ?, — ^ I are readily found with sufficient accuracy by an 

 rf/2 dt'' dt^J ^ ^ 



