588 HANSEN ON THE PERTURBATIONS OF BODIES IN 
orbits of a heavenly body, has in comparison with the other, real 
disadvantages, even while, through the labours of the greatest 
geometers of this century, it has been brought to a high degree 
of perfection. The author explains, in the Memoir read before 
the Academy, the most important of the disadvantages of this 
method ; here it is sufficient, not to occupy too much space, to 
remark, that in the deduction of the influence of the planets, or 
the perturbations, by means of mechanical quadratures, the cal- 
culations never come to an end, since it is necessary to repeat 
the same afresh from one period to another, to be able to con- 
nect the observations with each other; while, by the other me- 
thod, if the perturbations be computed once for all, we are by 
a short process enabled to connect with one another observa- 
tions lying close together, as well as those separated from each 
other by a long interval of time. 
All that we now haye to assist us in the solution of the pro- § 
blem in question is contained in the celebrated treatise of Gauss, § 
entitled Determinatio Attractionis, &c., in which is given an — 
elegant method by which can be calculated the secular variations, 
however great be the elliptic eccentricity and the inclination of 
the orbit. Although much is gained by this, yet it cannot be 
denied that a considerable gap yet remains to be filled up; for, 
in this treatise, we are not furnished with means of calculating 
the periodical terms, which are, in number and frequently also in 
magnitude, far more considerable than the secular variations. 
The author remarks upon this head, that, theoretically con- 
sidered, the perturbations can be calculated by means of the 
method given in his Essay that received the prize of the Royal — 
Academy in the year 1830, however great may be the elliptic — 
eccentricity and the inclination, for it is proved that the resulting ~ 
series are also in this case convergent. But the result arrived 
at in this way is practically useless, for, the convergence with 
large eccentricities being very small, it will consist of thou-— 
sands of terms. On the contrary, the method which the author ~ 
gives in the treatise read before the Academy, leads, at least in if 
the cases in which he has already applied it, to rapidly conver- — 
ging series ; and this can be relied on in it, that in all other cases — 
it gives the convergence of the serie: as great, or very nearly as 
great, as the nature of the circumstances admits of. It is indeed — 
evident in itself that the convergence cannot be the same in all 
cases. The method divides into two cases, accordingly as the 
