446 Scientific Intelligence. 



239th page of the memoir already cited, that this uncertainty might 

 be much greater, and amount even to 10" or 12", as is the case in the 

 tables of Saturn and Jupiter ', and as I have exhibited above. 



When we re-calculate the limits of the semi-axis major with other un- 

 certainties than 5'' in the data, we see that it is a great error to suppose 

 the extent of the limits of the semi-axis major varies proportionally to 

 the uncertainty of the data. It varies much more rapidly. Thus, al- 

 though for 5" of uncertainty in the data we find an interval of 2*86 be- 

 tween the limits of the semi-axis major, this interval so diminishes with 

 the uncertainty of the data, that when we reduce it to one half, we find 

 no value of the semi-axis major whatever that can satisfy the question. 

 And on the contrary when we carry the uncertainty of the modern data 

 above 5" we see that the lower and higher limits of the major axis change 

 rapidly, and leave a great latitude in the choice of this auxiliary, 

 " Without doubt," says Sir J. Herschel, " we may take for the starting 

 point any value for the semi-axis major comprised between 30 and 38," 

 It is not necessary to consider the period of revolution separately, 

 since it depends solely on the semi-axis major. But it is proper to con- 

 sider here another difficulty which has been supposed to exist in the 

 fact, that with the semi-axis major 30*20 the period of revolution of 

 Neptune should be very nearly double that of Uranus; a circumstance 

 which would introduce in the theories of these two planets, irregulari- 

 ties of considerable magnitude. It is a difficulty only of form, says 

 Sir J. Herschel with propriety, in his letter. And in fact so far from 

 these great irregularities being an embarrassment, we may neglect them 

 for a period of time equal to that which I have considered. I have 

 given the reason on page 157 of my memoir. In the inequalities of 

 the form A sin («/-H0> « is a very small angle, so that in the limits 

 wherein the time is comprised, we may substitute sin (ut-\-(l) for a pro- 

 gressive convergent series, according to the powers of the times, and 

 limited to the two first terms. Now these terms are confounded with 

 the mean longitude of the elliptical part of the movement of the dis- 

 turbed body, and may therefore be neglected in the calculation of the 

 perturbations. 



It is a serious error to suppose that the absolute values of the pertur- 

 bations, at a given epoch, can serve to determine the position of the 

 troubling body which produces them. We can only make use of the 

 variations which these perturbations experience, with the times, and yet, 

 must reject the part proportional to the times. The analytical formulae 

 which we employ to represent the position of the troubling body, ought 

 to satisfy only the condition of furnishing, during the interval of time 

 in which the perturbations are sensible, the same second differences of 

 the perturbations, that are obtained by quadratures, when we know 

 beforehand the geometrical situations of the disturbing body and its 

 mass. 



VII. Mass. — The mass of Neptune deduced from the observation of 

 its satellite is, according to Mr. Struve, 0*65 of the mass which results 

 from my theory; the extremest valuations reduce this down to 0*52 of 

 this mass. 



Now my theory of limits shows that an uncertainty of only 5" in 

 the modern data, allows of the adoption of a mass more than twice as 



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