182 THE ORBIT OF URANUS. 



e (in sec), 9678".69 



»i, 15425.752 



logrt, 1.2829072 



logr^i, 1.2831044 



Log rti includes, as before, the constant term in the perturbations of the logarithm 

 of the radius vector which, with the corrected mass of Neptune, is -[-•0001972. 



To iind the corresponding elements at any other epoch, the following secular 

 and long-period perturbations are to be applied. Those produced by Neptune are 

 derived from the expressions in Chapter III by correcting them for the new mass 

 of Neptune, and for the change in the value of the small divisor In — n produced 

 by the correction of the elements of Urauns. The logarithms of the factors for 

 correction are, 



Correction of mass of Neptune 9.93598 



Correction of divisor 0.00051 



Log. foctor for 11 9.93496 



Log. factor for.^e, ^7t, h-o 9.93547 



Including the perturbations of the second order just found, we have, by putting 



N=2l'—(j, 



= 113^ 30' 46".0 + 8° 26' bV'.^T, 



hi = (— 2850".41 + 0".32r) sin N^ (387".67 — 6".377') cos N 

 + 112.72sin2.V — 47.28 cos 2iV 



— 7. 72 sin 3i^ + 4.33 cos 3iV^ 

 -f- 0.55 sin 4i^ — 0.46 cos 4iVr 

 _ .037'^ — 83".78r+ 2811".41. 



hh = — 412".18 sin iV+ (14".03 — 0".867') cos N 

 + 29.20sin2iV— 6 .09 cos 2i^ 



— 3.11 sin 3.V+ 1.19 cos 3.V 

 -f . 28 sin 4.Y — 0.1 3 cos 4iV 

 -1- 14.76^ +398.33 



hh= — 4U".53 cos N— (13".65 — 0".867') sin N 

 + 29.33cos2A^+ 6 .17 sin 2iV 



— 3.12cos3iV^— 1.21 sin 3iV 

 + 0.29 cos 4iV^+ 0.13 sin 4iV^ 

 _ 5.453^ —124.72 



5» (in units of the 7th place of decimals). 



= 1963 cos iV+103sin N 



— 171cos2iV— 67sin2i\^ 



+ 15cos3iV+ 6sin3iV^ 



+ 511.0. 



The perturbations of ehn and h are here replaced by those of h and Tc, defined 



by the equations 



h =z e sin {~t — tTq) 

 i = e cos [n — Tto) 



