109 



Jupiter and Saturn, it was only necessary to take terms of the 3rd 

 order of smallness, and the multiplier by which the terms are aug- 

 mented has 30 2 instead of 240 2 for its factor. 



In the present state of physical astronomy methods exist by which 

 the results of the law of universal gravitation in the planetary system 

 may be obtained to any degree of accuracy, by calculating in suc- 

 cession the terms of successive orders of minuteness, the order being 

 estimated according to the powers and products of the excentricities 

 of the orbits. But it is well known, that in the actual application of 

 these methods, the number of the terms arising from the combination 

 of the several series which occur, and the complexity of the operations 

 by which the coefficients of these terms are to be deduced, increase 

 so rapidly in passing beyond the lower orders of inequalities, that 

 the calculation is difficult and laborious. 



The numerical calculation of a perturbation depending on the 5th 

 powers of the excentricities has not been executed, so far as we are 

 aware, except in the case of the great inequality of Jupiter and Sa- 

 turn, where, as Laplace states {Mecanique Celeste, p. ii. liv. 6. 9°), this 

 labour, " penible par son excessive longueur," has been performed by 

 Burckhardt with a scrupulous attention. And no calculation of a 

 new inequality of a high order, requiring to be placed in the pla- 

 netary tables, with a new r argument, has been published since that 

 of the great inequality by Laplace in 1784. 



One of the main parts of the labour of such calculations consists 

 in obtaining the successive terms of a certain quantity on which the 

 perturbing forces depend, and which in the Mecanique Celeste is 

 called R. This quantity is a function of the positions of the two 

 planets which affect each other, and involves the reciprocal of the 

 distance of the two bodies. It is to be expanded according to the 

 powers and products of the excentricities and inclination of the two 

 orbits, its successive terms having as factors the cosines of certain 

 angles, all of which increase proportionally to the time. 



It may be expanded by Taylor's theorem applied to several va- 

 riables, according to powers of the excentricities, and f~, f being the 

 sine of half the inclination*. 



In this expansion the coefficients of the cosines of the different 

 arguments are functions of certain quantities A or b (according to the 

 notation of the Mecanique Celeste), and of the partial differential co- 

 efficients of these quantities with regard to a and a', the radii of the 

 orbits ; admitting however of reduction so as to contain the differen- 

 tial coefficients with regard to one of these quantities only. 



The quantity b is a function of and has several values di- 

 stinguished by different indices : these are connected by certain 

 well-known equations of condition. The author of the present me- 

 moir obtains the development of R in terms of quantities C, sym- 



* Laplace uses s, the tangent of the inclination. Burckhardt expresses 

 himself to be in great doubt what function of the angle it is best to take. 



i 2 



