THE ORBIT OF NEPTUNE. 19 



Such arc the formula} by which we shall proceed to compute the perturbations 

 of Neptune by Jupiter, Saturn, and Uranus. 



It will be noticed that the coefficient of ft vanishes identically in the last de- 

 velopments. I have not completely investigated this law, but it seems to arise 

 from the circumstance that that portion of the perturbation in question which 

 proceeds from the change in the origin of co-ordinates is independent of (i, while 

 that portion which is caused by the modified attraction of the Sun is of the order 

 of magnitude ^. It furnishes a yet more valuable check than the last on the 

 developments. 



11. Allusion has already been made to the complications introduced into the 

 theory of Neptune by the near approach of its mean motion to double that of 

 Uranus, and the consequent oscillation of all the elements of its orbit in a cycle 

 of 4300 years of duration. In order to construct a dynamical theory which should 

 be correct within a tenth of a second through the whole of one of these cycles, it 

 would be necessary to include many terms dependent on the second, and perhaps 

 some dependent on the third power of the masses of the disturbing planets. 



If this task were accomplished, the necessary uncertainty in the mass of Uranus 

 and the elements of Neptune would destroy all the value of the theory. A change 

 of one-tenth in the mass of Uranus would produce a change of 200" in the co- 

 efficient of the perturbation of the mean longitude. The mean motions of Walker 

 and Kowalski being each about 8" in error, the place of the planet from this cause 

 alone would be in error by nearly 10 at the end of a cycle. 



After much careful consideration of different ways of relieving the theories of 

 Uranus and Neptune from the complexities introduced by the large perturbations 

 referred to, I finally determined to develop them not as perturbations of the co- 

 ordinates, but of the elements. It will readily be seen that if the eccentricity or 

 perihelion is greater than the mean during several revolutions of the planet, there 

 will be a perturbation in the radius vector and longitude having nearly the same 

 period with the revolution of the planet, although the latter may really scarcely 

 wander from a true elliptic orbit during an entire revolution. In such a case it 

 is clearly best, in constructing a theory designed to remain of the highest degree 

 of exactness for only a few centuries, to take not the mean values of the elements, 

 but their values at a particular epoch during the time the theory is expected to 

 be used. 



In doing this, we shall be treating the change in the elements in the same way 

 that the secular variations are usually treated. These variations are really 

 periodic, and in a perfect theory would have to be treated as such. But the 

 elliptic elements on which all our planetary theories are founded are not mean 

 elements, but elements brought up by secular variation to the epoch 1800 or 1850. 



Thus, our perturbations of the elements will be of the form 



01 IT 



Sa zz c -f aj + 2a 2 {kt-^-s}, 

 cos 



iji which a' is the secular variation proper, k a small coefficient equal to 2 ri n 

 or its multiples, and c a constant added to the integral, of such value as to make 

 &a vanish at the epoch 1850. 



