THE ORBIT OF URANUS. 13 



added to v^, \;/, etc., to make the true values of v, v', etc., whether perturbations or 

 corrections of the elemcuts. We shall then have 



<5v„ dVo dpo ♦^ ' dp'o ^^ 6y^ ' 



^-^T--^v,^-^" + -c^v-„<5v';^^ + -^v,5p7^P+ ^^v„<?p-^P+Tv,^^^ (11) 



The value of Z>'t72 may be found either by equation (2), or by differentiating with 

 respect to the time as introduced by the co-ordinates of the disturbed planet. 

 When quantities of the first order only are considered the latter operation is very 

 simple, but it is different when terms of the second order come in, because the true 

 longitude of the planet is then expressed in terms not only of its own mean longi- 

 tude, but also of the mean longitude of all the disturbing planets. The result can 

 still be obtained in the same way by separating all the mean longitudes introduced 

 by the co-ordinates of the disturbed planet from those introduced by the co-ordinates 

 of the other until after the differentiation relatively to t'. 



Let us now resume the equation (4), representing its second member by fi Q, so 

 that it becomes 



'^li-m.^tLO^palr.^Sp^^.Q (12) 



tit / Q 



where 



Q = 2SD\Rat + f^^ _ 1 ^ill!i'M + 1 ¥ 



By the operations already given Q has become a known function of the time. 



It is well known that the integration of (12) may be effected by finding two 

 values of rj^8p which satisfy this equation when the second member is neglected, 

 or, in other words, by finding two variables x and y which satisfy the equations 



fPx ^<(1 + m) 

 when the required integral is 



The above differential equations are satisfied by the rectangular co-ordinates of the 

 planet in its assumed elliptic orbit. The position of the axes of co-ordinates being 

 arbitrary we shall take the line of apsides for the axis of X, the perihelion being 

 on the positive side. If we put 



e^ = sin 4,, 

 we have 



dy dx / — - — i.i(l -\- m) cos 4* 



X 



y jf = Va^ (1 + m) cos 4- 



dt ^ dt~ " "'" ^" ^ ""' """ ^ ~ an 



