178 THE ORBIT OF UK AN US. 



CHAPTER VIII. 



COMPLETION AND ARRANGEMENT OF THE THEORY TO FIT IT FOR 



PERMANENT USE. 



IN the preceding discussions the terms of the second order due to the action of 

 Neptune have been neglected, the elements of Uranus and Neptune being so chosen 

 that these terms can scarcely become sensible within a century of the epoch. But 

 this very choice will make them larger in the course of centuries than if mean 

 elements had been chosen. They will be most sensible in the case of the great 

 inequality of 4300 years between Uranus and Neptune, an inequality which will 

 make centuries of observation necessary to an accurate determination of the mean 

 elements of the two orbits. The uncertainty arising from the great inequality is 

 probably of the same order of magnitude with the omitted terms of the second 

 order, and, such being the case, the theory would really be made but little more 

 accurate by the addition of those terms. I conceive, however, that the theory will 

 be made much more satisfactory by the computation of at least the largest of the 

 terms in question, if only to arrive at a certain determination of their order of 

 magnitude, and of their effect on the planet during the period in which it has been 

 observed. 



The term in question, being of very long period, may be most advantageously 

 treated by the method of variation of elements, more especially as it has in the 

 theory been already treated as such a perturbation. The largest of the pertur- 

 bations in question are those of the mean longitude which are multiplied by the 

 square of the integrating factor v, which is nearly 51, but which also contain the 

 eccentricities as factors, and those of the eccentricity and perihelion which arc 

 independent of the eccentricities, but are multiplied by only the first power of v. 

 These terms will probably comprise nearly or quite nine-tenths of those arising 

 from the term of long period. 



Let us begin with the perturbations of mean longitude. These are given by the 

 integration of the equation 



-*= 3m'an 2 \ek L sin (11 I n) -j- e'k 2 sin (21 I n'}} 



&! and 7c 2 being functions of the ratio of the mean distances, or a. If we integrate 

 this equation, supposing all the quantities in the second member except T and I to 

 be constant, and these two to be of the form nt -\- e, n and e being constants, we 

 shall reproduce the principal term of long period already found. But in the 

 second approximation we must suppose all the elements variable. It is not, how- 

 ever, necessary to take into account the variations of a, n, and 7c, because these are 



