22 Rev. B. Bronwin on the Determination of 



Let r x and v x be the same functions of the new time £, and 

 the constants h v e v ir^ s which r and u are, when there is no 

 disturbing force, of the time / and the constants h , e Qi 7r , and 

 « . Also let v = v l + 6t. We shall have 



-- = r~2{ 1 + *i cos (°i-' r o)}- 



r \ n i 

 We shall not with M. Hansen find the log of r, and there- 

 fore shall make — m — + p- Substituting this value, we shall 

 easily find 



P = ^(n^-h^ + ^(n^-ni) COS ^-' V o) 

 e n St 



h 2 



"0 



sin (u, — w ) + P. 



Whence p is of the order of the disturbing force, and it has 

 the advantage of requiring only one integration. 

 In virtue of the supposed equation 



we have 



dv dvy p _dv x d^ p _ h l d% > g 



d~t~"dt ' + ~d\Tt* r*dt + 



This value, substituted in the known equation 



gives 



Of the four quantities h , h v e oi e v two are to be found in 

 terms of the others, which will be arbitraries of the theory ; 

 and the mode of determining them will be obvious after the 

 development is effected. 



Putting <J> for the latitude, i for the inclination, 3 and for 

 the longitude of the node on the plane of the orbit and on 

 the fixed plane, we have 



3 = / cos i d 0, sin <J> = sin i sin (o — S) 



= sin i (cos ^ sin u — sin •& cos u), 



sin? cos •& = sin/ cos$ + / (cosz'cosSd? — sin? sin 3 ^3), 



