172 
SECOND REPORT — 1832 . 
moires , 1816. The point to which greatest attention is paid 
is the variation of the axis major. Laplace had previously 
shown that it contained no permanent terms to the third powers 
of eccentricity, &c. and the first order of the disturbing force : 
Lagrange had exten&Ied this to all terms of the first order of 
the disturbing force : Poisson now extended it to the second 
order of the disturbing force, as far as fourth powers of eccen¬ 
tricity, &c.; and Lagrange showed that the same theorem is 
true generally to the second order of forces, whether we con¬ 
sider the perturbing body to be itself liable to perturbation, or 
not. 
In the Gottingen Transactions 1816-1818, Gauss investi¬ 
gated the secular variations of the elements, supposing the dis¬ 
turbing body extended over the line of its orbit, the proportion 
of the thicknesses at different points being the same as that of 
the time actually occupied in describing a given length. The 
ingenuity of the transformations, &c. deserves notice, but the 
theory of perturbations has gained nothing. 
In the Memoirs of the Astronomical Society , vol. 2, M. Plana 
made some remarks on the correctness, in point of form, of 
Laplace’s investigation relative to the constant alteration in the 
axis major, and on the accuracy of his results as to the effect of 
the attraction of the stars. In the Conn, des Temps 1829, La¬ 
place made some alterations in his investigation of the latter. 
In Nos. 166, 167, 168, and 179 of the Astronomische Nach - 
richten. Hansen has presented the theory (with reference to 
practical applications,) in a form that well deserves attention. 
Instead of determining the true longitude by means of the 
usual elements, all which (including the mean longitude cor¬ 
responding to any given instant,) are variable, he assumes that 
the true longitude shall be determined by the usual expression 
for longitude applied to invariable elements, the mean longitude 
only (at any fixed epoch) being considered variable. He as¬ 
sumes also that the true radius vector shall be determined by 
applying the usual formula for the elliptic radius vector to the 
same invariable elements and variable epoch of mean longitude, 
and adding to this expression certain variable terms. This 
method was probably suggested to its author by the observa¬ 
tion that, in the great inequalities of long period, the variation 
of epoch is much more important than the other variations. At 
all events, it is a form particularly well adapted to the construc¬ 
tion of astronomical Tables, and the more so as Hansen found, 
in application, that the convergence of the terms in this method, 
especially for the higher orders of the disturbing force, was 
more rapid than in any other. Laplace’s or Lagrange’s expres- 
