1 THE THEORY OF THE LONG INEQUALITY 



rn' s n 3 a?C 3 e , „. IdP' 



% e . ,AdP' . dP 1 



„ -cos (-ar - -mil—- sinX — cosX>, 



e [ de ae J 



to sin 1 



mm'nriaa , . , 



— : — =r- CjJfi ) sin -jsr sin X - cos •zsr cos X J , 



a) sin 1 



= + 0"-017 sin X + 0"-081 cos X, 



which may be rejected. We may also reject the corresponding terms of eXw, 5V, e'^'sr', all 

 of which are very small. 



For the effect of Saturn on Uranus, the formula is 



„ TOim'wVCs' e y tdP" dP 1 



Co 2 sin 1 e [de de 



and a similar formula will give the effect due to the action of Jupiter : but these terms may 

 be neglected ; for all the coefficients are less than half a second. 



57. The terms calculated in the preceding articles seem to be all of those due to the 



square of the disturbing force, which are of sufficient magnitude to be retained. But there 



are other larger terms of two dimensions in the masses, depending upon the constant corrections 



which must be applied to the values of the elements used in making the first approximation. 



Let Ae, Aw, Ae', A-ar' be the corrections which must be applied to the values of e, •&, e, •&■', 



used in the first section, so that e + Ae, &c. are accurately equal to the constants introduced 



de 

 in integrating the equations for — , &c. : then, if e, ■&, &c. were accurately the elements of the 



instantaneous ellipse at the epoch t = 0, Ae would be given by the equation 



Ae= -2(tSe)* = , 

 2 (cSe) being the sum of all the perturbations of the eccentricity produced by the action of all 

 the planets: and similarly for Atst, &c. We shall calculate the values of these corrections only 

 in so far as they depend upon the long inequality. 



Since e and •ar (as deduced from observation) are given for the epoch of 1801, we must put 

 t = — (46 x 365 +11) days in the expressions for $e and ci-ar, in calculating the values of Ae 

 and A"5T. If, then, X"= the value of X for Jan. 1, 1801, = 83° 12' 35", 



Ae = - 85"-105 sin X" - 408"'350 cos X" 



- 6"-989 sin 2X" - 32"'541 cos 2X" 



- 0"-759 sin 3X" - 3" -21 9 cos 3X" 

 = - 100"-967. 



eA-ar = + 409"-040 sin X" - 84"*810 cos X" 



+ 32"-498 sin 2X" - 6"*988 cos 2X" 



+ 3"-219 sin 3X" - 0"-759 cos 3X" 

 = + 407"'811. 



And if X' = the value of X at time t = 0, 



= 78° 28" 36"'4. 



