330 ON THE MOTION OF 



planets and of the inclinations of their orbits 43 , and that of 

 Laplace which is extended so as to include all their variations, 

 whether periodical or secular 44 , are alike remarkable for the 

 analytical treasures they contain and the singular success with 

 which this purely intellectual apparatus is made to declare 

 the minutest and most prolonged of the celestial oscillations. 



In 1788 Lagrange published his Analytical Mechanics. 

 The first paragraph of the fifth section of the first edition 

 of this work is a masterly investigation of the small oscilla- 

 tory motions of any system of bodies round the places of their 

 rest. The great generality of this solution, along with its use- 

 ful applications and manageable formulas, render it altogether 

 one of the most important contributions ever made by mathe- 

 matics to mechanical philosophy". The equilibrium posi- 

 tions of the elements are supposed, in Lagrange's dissertation, 

 to be determinate and unique ; that is, the system is supposed 

 such that it cannot change its position without departing from 

 a state of equilibrium. It is manifest however that in a large 

 variety of cases, a system of material points may have a 

 range, more or less extensive, in any part of which it will 

 remain at rest. If the analysis of Lagrange had been made 

 to comprehend, as far as that is practicable, the motions of a 

 system in the immediate neighbourhood of its range of equi- 

 librium, the subject would have been exhausted, and the limits 

 of the science in no small degree enlarged. 



After Huyghens and James Bernoulli had completed the 



43 Recherches sur les equations seculaires des mouvemens des nceuds et des 

 inclinaisons des Orbites des Planetes. Mem. Acad. Paris, 1774, p. 117. This 

 paper, though of posterior date, is quoted by Laplace in the memoir following : — 



** Recherches sur le calcul integral et sur le systeme du mondc. Mem. Acad. 

 Paris, 1772. P. ii. p. 293. 



* s It may be well to mention for the benefit of those who may find it useful to 

 employ these formulas, that by some oversight on the part of Lagrange the values 

 of all the brackelted coefficients in the final differential equations are deficient in 

 all the quantities which arise from having regard to the terms of the second order 

 in the devclopiuenis of the coordinates of the elements. In the American Journal 

 of Science and Arts for July — Sept. 182G, p. 398, I have given the terms ncces- 



j to complete the values of these coefficients, with some remarks as to the best 

 form of the function which expresses the finite action of the impressed forces on 

 any one of the corpuscles of the system. 



