T1IK OKI! IT OF URANUS. 



65 



l\rt iii-lit inn* of Saturn. 

 The perturbations produced by Jupiter will be taken from the exhaustive pri/.e 

 memoir of llaiix-u. 1 As the perturbations required are those of the co-ordinates, 

 it will be necessary to transform those of llansen into the usual form, llauseu 

 gives the true anomaly r in the form 



v = g -\- nlz -j- r, sin (<j -f nfz) -(- e, sin > (</ -}- n&z) -J- etc., 



<>,, r 2 , etc., being the coefficients of the multiples of the mean anomaly in the usual 

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



of /'::, 



lv = n'z (1 -j- e, cos <7 -j- 3e, cos 2# -\- etc.). 



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

 ^ ;/ v . in this expression. In the same way, we have for the perturbations of log r, 



fy = fy a -j- nlz (e w sin g -\- **> sin 2 g + etc.) 



fy, being Ilansen's perturbation, and e (t} the negative coefficient of cos iy in the 

 development of the elliptic log r. 



Hansen having adopted -jufj-.-s as the mass of Jupiter, it will be necessary to 

 multiply his perturbations by 1.0'216 to reduce them to Bessel'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 method, and are given in the following table. In the table /' is the mean 



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



of Saturn. fy> is the perturbation of the Naperian logarithm, in units of the 



nth place of decimals. 



1 Untersachungen iiber die gegcnseitigcn Stb'rungcn dee Jupiters und Saturns. Von P. A. Han- 

 ten. Berlin, 1831. 



9 April, 1873. 



