380 Royal Astronomical Society. 



"This expression proceeds in an infinite series of terms, both in 

 the vertical and horizontal directions. It, however, converges very 

 quickly for any given values of t and i 1 , so that in general only a 

 very small number of coefficients require to be taken into considera- 

 tion in order to determine the coefficient of the integral ; and since, 

 for the sine and cosine, the expression is merely a numerical quan- 

 tity, when fx and /u' are given, so is the calculation for the very small 

 number of values of t, when i' is constant, by no means difficult. 



" Dr. Briinnowhas succeeded in avoiding the indirect solution by 

 adding to the equation relative to the undisturbed orbit, viz. 



the equation for the disturbed orbit, viz. 



A ^=v(l_l) +x X +y Y + *Z + Sx.± + Y d J- + Z.±) . 

 2 d? \r a) y JV <# <ft dt J ' • 



Whence, by comparison, there result the following rigorous equa- 

 tions : — 



^dHf^_^_^ =xX+yY+zZ + ¥iX(lx+Ydy+Zd2h 

 <F(.r- 



and let us collect together the squares and products of <>r, £, r\ and 

 £. We hence deduce these four equations : — 



dd(r°Zr) 

 df 2 + 



+ 3^(rVr) 



