372 HISTORY OF MECHANICS. 



the mode of obtaining the secular variations of the elements. Poisson 

 and Lagrange proved the invariability of the major axes of the orbits, 

 as far as the second order of the perturbing forces. Various other 

 authors have since labored at this subject. Burckhardt, in 1808, ex- 

 tended the perturbing function as far as the sixth order of the eccen- 

 tricities. Gauss, Hansen, and Bessel, Ivory, MM. Lubbock, Plana, 

 Pontecoulant, and Airy, have, at different periods up to the present 

 time, either extended or illustrated some particular part of the theory, 

 or applied it to special cases ; as in the instance of Professor Airy's 

 calculation of an inequality of Venus and the earth, of which the period 

 is 240 years. The approximation of the Moon's motions has been 

 pushed to an almost incredible extent by M. Damoiseau, and, finally, 

 Plana has once more attempted to present, in a single work (three 

 thick quarto volumes), all that has hitherto been executed with regard 

 to the theory of the Moon. 



I give only the leading points of the progress of analytical dynamics. 

 Hence I have not spoken in detail of the theory of the Satellites of 

 Jupiter, a subject on which Lagrange gained a prize for a Memoir, in 

 1766, and in which Laplace discovered some most curious properties 

 in 1784. Still less have I referred to the purely speculative question 

 of Tautochronous Curves in a resisting medium, though it was a sub- 

 ject of the labors of Bernoulli, Euler, Fontaine, D'Alembert, Lagrange, 

 and Laplace. The reader will rightly suppose that many other curious 

 investigations are passed over in utter silence. 



[2d Ed.] [Although the analytical calculations of the great mathe- 

 maticians of the last century had determined, in a demonstrative man- 

 ner, a vast series of inequalities to which the motions of the sun, moon, 

 and planets were subject in virtue of their mutual attraction, there 

 were still unsatisfactory points in the solutions thus given of the great 

 mechanical problems suggested by the System of the Universe. One 

 of these points was the want of any evident mechanical significance in 

 the successive members of these series. Lindenau relates that Lagrange, 

 near the end of his life, expressed his sorrow that the methods of ap- 

 proximation employed in Physical Astronomy rested on arbitrary pro- t 

 cesses, and not on any insight into the results of mechanical action. 

 But something was subsequently done to remove the ground of this 

 complaint. In 1818, Gauss pointed out that secular equations may be 

 conceived to, result from the disturbing body being distributed along 

 its orbit so as to form a ring, and thus made the result conceivable 

 more distinctly than as a mere result of calculation. And it appears 



