2 TIIEORBITOFNEPTUNE. 



Ne2:)time. The longitude of perihelion referred to the mean Equinox of Jan. 1, 

 1847, eccentricity, and mean daily motion were as follows : 



7t = 48°2r2".93 

 e - .00857741. 

 n — 21".55448. 



This mean motion rendered it certain that the supposed relation between the 

 mean motions of the planets Ui'anus and Neptune had no foundation in fact. 

 Professor Peirce thereupon revised his theory, and published the new perturb- 

 ations in the Proceedings of the American Academy, Vol. I, p. 286. 



The near approach to commensurability of the mean motions renders the 

 general theory of the mutual action of Uranus and Neptune extremely complex. 

 Twice the mean motion of the latter exceeds that of the former by only 320" 

 according to Walker, or 304" according to my first revision of his elements. The 

 terms in the perturbations which contain this very small quantity as a divisor 

 will, therefore, be very large. Considered as perturbations of the elements, their 

 period will be more than ^^ years. We have an analogous instance in the 900 

 year equation of Jupiter and Saturn. But in the latter case the perturbations 

 of the mean motion are of the third order with respect to the eccentricities and 

 inclinations, while in Uranus and Neptune they are of the first order. From this 

 circumstance it happens that, notwithstanding the smaller masses of the dis- 

 ■turbing planets, the perturbation of the mean motion is as great in the case of 

 the planets in question as in that of Jupiter and Saturn, and that of the other 

 elements enormously greater. In fact, the perihelion of Neptune oscillates through 

 a space of eiglit decrees in consequence of the terms in question. Such a perturb- 

 ation as this, four degrees on each side of the mean, is, I think, found nowhere 

 else in our system. Moreover, a change of 1" in the mean motion of the planet 

 will produce a change of nearly 2' in the coefficient of this perturbation. Any 

 attempt to determine its magnitude with accuracy will, therefore, be hopeless. 



But the difficulties connected with these terms can be avoided in the case of a 

 theory which is designed to be exact for a period of only a few centuries. Not- 

 withstanding the great magnitude of the general integrals of the perturbations, 

 if we take these integrals between limits not exceeding a coujjle of centuries, we 

 shall find them so small as not to involve serious difficulty. Their effect on the 

 co-ordinates can then be developed in powers of the time, and the values thus 

 obtained will not be subject to any uncertainty of moment. This is substantially 

 the course adopted by Professor Peii'ce. He says of the terms in question : 



" These coefficients will vary very sensibly by a change in the value of the 

 mean motion of Neptune, arising from a more accurate determination of its orbit. 

 But the principal effect of these terms can for a limited period, such as a century, 

 for instance, be included in the ordinary forms of elliptic motion, and the residual 

 portion will assume a secular for-m which is no more liable to change from a new 

 correction of tbe mean motion of Neptune than the other small coefficients of 

 the equations of perturbations." 



Accordingly, subducting from the terms in question a series of expressions 



