524? Royal Astronomical Society. 



method of varying the elliptic elements led to the sublime disco- 

 veries now alluded to, it is not without defects, which become parti- 

 cularly sensible in the numerical computations. Among these is 

 to be reckoned the length of the calculations which it renders ne- 

 cessary for two reasons ; first, because the,number of elements of an 

 orbit being twice the number of the coordinates which determine 

 the place of the body, the calculation of a much greater number of 

 quantities is required than by the first- mentioned method ; and, se- 

 condly, because when the perturbations of the elements have been 

 computed, there still remains the labour of substituting the altered 

 elements in the expressions of the coordinates derived from the el- 

 liptic motion, in order to obtain the disturbed coordinates and the 

 place of the body in its actual orbit. The principal defect of the 

 method, however, consists in this, that the coefficients of the dif- 

 ferent terms of the series which express the disturbed elliptic ele- 

 ments have, in general, much larger values than the corresponding 

 terms of the expressions of the disturbed coordinates which deter- 

 mine the position of the body, so that the series expressing the 

 disturbed elements converge slowly, even when they correspond 

 to small perturbations of coordinates. If we conceive, for example, 

 a system of forces of short period to disturb the curvature of an 

 orbit many times in a single revolution, it will be easy to see that 

 in each of these periods the elements of the orbit may have been 

 greatly altered, while the disturbance of coordinates (of the longi- 

 tude and radius vector, for example) may have been trifling. But 

 in order to obtain these small disturbances, it is necessary to pass 

 through the perturbations of the elements, which, relatively, are 

 very considerable, and of which the calculation is rendered laborious 

 by reason of the slow convergence of the series ; and this incon- 

 venience exists not merely in the case of the perturbations of the 

 first order with respect to the masses, but in a still greater degree 

 in the case of those of the second and of the higher orders. For 

 these reasons the calculation of the perturbations has hitherto 

 been in some respects imperfect and unsatisfactory ; the computer 

 always finding himself obliged to omit a number of the smaller terms 

 without having any assurance that the error resulting from the omis- 

 sion is insensible ; or, as M. Hansen has remarked, rather from a 

 sort of presentiment that the omitted terms have no appreciable in- 

 fluence, than from a mathematical demonstration of their influence 

 being insensible. 



It was with a view to remove these defects from the lunar and 

 planetary theories that M. Hansen undertook the series of remark- 

 able investigations which have appeared from time to time, during 

 a considerable number of years (partly in Professor Schumacher's 

 invaluable Repertory, the Astronomische Nachrichten, and partly in 

 two separate publications, — one on the perturbations of Jupiter and 

 Saturn, and the other on the lunar theory), for which the Council has 

 now awarded the Society's medal. His method of expressing the 

 perturbations is based on that of Lagrange ; but the modifications 

 which he has introduced are of an important kind, and lead, in fact, 



