THE ORBIT OF URANUS. 



65 



General Perturbations of Saturn. 

 The perturbations produced by Jupiter will be taken from the exhaustive prize 

 memoir of Hansen.^ As the perturbations required are those of the co-ordinates, 

 it will be necessary to transform tliose of Hansen into the usual form. Hansen 

 gives the true anomaly v in the form 



v =^ rj -\- nlz -\- e^ sin {(j -\- nh) -|- e. sin 2 {(j -\- nh) -)- etc., 



Cj, ^2, etc., being the coefficients of tlie multiples of the mean anomaly in the usual 

 development of the elliptic true anomaly. Whence, neglecting the second power 

 of n^^z, 



hi = nlz (1 -|- Ci cos g -\- le.^ cos 2^ -\- etc.). 



To make the development sufficiently rigorous it is only necessary to increase rj by 

 \nhz in this expression. In the same way, we have for the perturbutions of log ?•, 



^p ^ f^p,, -j- nlz (c<^' sin fj -\- e'-' sin 2 </ -j- etc.) 



i^po being Hansen's perturbation, and c<'* the negative coefficient of cos i(j in the 

 development of the elliptic log r. 



Hansen having adopted join. 5 ^^ the mass of Jupiter, it will be necessary to 

 multiply his perturbations by 1.0216 to reduce them to Bcssel's mass. Thus the 

 perturbations by Jupiter hereafter given have been obtained. 



The perturbations by Uranus and Neptune have been computed by the preceding 

 general metliod, and are given in the following table. In the table V is the mean 

 longitude of the disturbing planet, Uranus or Neptune, counted from the perihelion 

 of Saturn. 5p is the perturbation of the Napcrian logarithm, in units of the 

 seventh place of decimals. 



' Unt.ersuchungeii iiber die gegenseitigen Stbrungen des Jupiters und Saturns. Von P. A. Han- 

 sen. Berlin, 1831. 



9 April, 1873. 



