OF URANUS AND NEPTUNE. 



Ixi 



is at 6, reaches its maximum when A is at a, and then decreases, reaching its mean value 

 when A is at b' , and its minimum when A is at a. 



The same force causes the perihelion to recede in the arcs Aba , A'b'a, and to advance in 

 the arcs aA, a A'. The perihelion therefore is in its mean place when A coincides with it; 

 has accomplished its greatest recess when A has arrived at b ; has returned to its mean place 

 when A is at a ; attains its greatest advance when A is at b' ; and again returns to its mean 

 place at the end of the revolution of A. 



The normal force produces perturbations of the eccentricity and perihelion of the same 

 sign as those produced by the tangential force, and which respectively reach their maxima and 

 minima simultaneously with them*. 



70. Changes in the eccentricity and longitude of perihelion necessitate a change in the 

 mean longitude, which the planet is supposed to have at the epoch ; but this element is 

 not affected in any other way. The true longitude of the planet at any time t depends 

 upon the mean longitude at epoch, the mean motion during the interval from the epoch 

 up to the time t, and the eccentricity and longitude of perihelion of the instantaneous 

 ellipse at that time. Suppose the interval between the epoch and the time t to be divided 

 into a great number of very small portions, and suppose the planet to move during each 

 of these portions of time in the instantaneous ellipse which belongs to the beginning of 

 that portion. By indefinitely diminishing the length of these divisions of time, and increasing 

 their number, we may make this hypothetical motion of the planet approximate as nearly as 

 we please to the real motion. Now the mean motion during the time t which has elapsed 

 since the epoch, may be estimated in either of two ways. We may either put it equal 

 to the mean angular velocity in the instantaneous ellipse at the time t, multiplied by that 

 time (nt) : or we may multiply the mean angular velocity of the instantaneous ellipse at 

 the beginning of each small portion of time, by the length of that portion, and take the 

 sum of all these products from the epoch up to the instant in question (fndt). We shall 

 suppose it to be estimated in the latter way. This sum, then, at the time t, depends upon 

 the values of the mean angular velocity during all the interval from the epoch up to the time t, 

 but not upon its value at that instant. Consider now the subdivision of time immediately preced- 

 ing the completion of the time t. The mean longitude at epoch is so adjusted, that, with 

 the eccentricity and perihelion belonging to the beginning of this portion of time, and with 

 the mean motion calculated for each instant in the above manner, the planet may have its 

 true longitude at every instant, during this small interval of time, and therefore both at the 

 beginning and end of it. At the time t, the planet proceeds to move in a different ellipse ; 



dvs 



dR dd dR dr 



dd dvs dr dvs' 



— =-2ecos/3, r-=-oesin/: 

 dvs dvs 



therefore taking the circular part of the disturbing forces only, 



dR 



dR 



dR 



•7- = - -=- 2« cos B - — ae sin B, 



dvs iltj, r da 



= - 2m'^ g sin ( 2<£ - 2tf>' ) . 2e cos /3 



.dJ* 

 +m'-—coi(2<p- 

 da 



(>'). ae sin/3, 



= - / 2m' A 2 + 1 m'a —~ I esin (\ + vs), &c, 



= -m'M\e sin (X + nr), &c. ; 



de na dR , „ _. _ 



.•. — = — -p- = -m naM , sin (\ + vs), 

 dt e dvs 



which agrees with the formula of Art. (16). The principal 

 part of the perturbation of the eccentricity therefore depends 

 upon the circular disturbing forces ; and the same is true of the 

 longitude of perihelion. 



