THE ORBIT OF UllAXUS. 



81 



Action of Jupiter, 

 Action of Saturu, 

 Action of Neptune, 



3Ii(, = -i-isrcosg 



+98 2' cos /; 

 + GTcosg 



+17' cos 2g 

 +7 2' cos 2g 



The secular variatioix of the longitude of the perihelion is 



Action of Jupiter, 

 Action of Saturn, 

 Action of Neptune, 

 Total. 



+122, ir 



+ 118.4 r 



+ 51. ir 



««=+291.6r 



The effect of this secular variation on the longitude and radius vector is 



Action of Jupiter, 

 Action of Saturn, 

 Action of Neptune, 

 Total, 



6v = 



— 11. 40 r cos (7 

 — ll.llTcossr 

 — 4. 80 7' cos gr 

 — 27.37 T cosy 



-O.G712'cos 2j 

 -O.CSlTcos 2g 

 -0.281 2' cos 2g 

 -1 603 r cos 2g 



—0.047 2' cos 3g 

 — 0.039 r cos 3(7 

 — O.Olfircos Sg 

 —0.102 2' cos 3g 



Action of Jupiter, 

 Action of Saturn, 

 Action of Neptune, 



Mip = 



—120 2' sin 3 

 — 117 2' sin g 

 — 50 2' sin g 



—8 T sin 2g 

 —8 T sin 2g 

 —3Tshi 2g 



For the purpose of conveniently tabulating the perturbations, we shall express 

 them in a form similar to tliat adopted in the theory of Neptune. Let us select, 

 from the terms of the periodic perturbations produced by any planet, all those in 

 which the difference between the indices i and i' is the same. For example, in 

 the perturbations of the longitude produced by Jupiter, let us consider the terms 



^» = + 1.269 sin ( — /) 



— 3.495 sin (2^ — I) 

 -f 1.182 sin ( (7 — 20 

 + 0.074sin(35r— 2/) 



— 0.005 sin (2(7 — 30 

 + 0.011 sin (4j — 30 



These terms may be expressed in the form 



+ 0.002 cos ( — 



— 0.092 cos (2^7 — 

 + 0.515 cos ( j7 — 20 



— 0.005 cos (3f7 — 2/) 



— 0.001 cos (4^ — 30 



^v = sin ff X 



+ 0.094 sin {g — T) 

 + 0.520 sin 2((7 — 



I —2.226 sin (g — l) 



+ cos (/ X ■{ _|_ 1.256 siii 2{!j — I) 



I + 0.006 sin 3(^7 — 



11 May, 1873. 



— 4.764 cos (g — 



— 1.108cos2(<^ — 

 + 0.016 cos 3(g — 



— 0.090 cos (g — I) 

 + 0.510 cos 2(i7 — 



