522 Royal Astronomical Society. 



other, and the amount hy which the elements of their fluctuating 

 orbits deviate from their average conditions ; — to express in formula? 

 the state of the system and the position in space of every body be- 

 longing to it at any given instant in past or future duration ; and, 

 finally, to convert his formulae into numericalt ables, for the uses of 

 navigation and the other important purposes to which astronomy is 

 subservient. 



It is for researches in this second department of our science, un- 

 doubtedly the most arduous and difficult of the two, that your Council 

 have awarded the Society's Gold Medal for the present year to 

 Professor Hansen, the Director of the Observatory at Seeberg, and, 

 according to annual custom, the duty devolves on me of stating to 

 you the grounds of their decision. The subject is not very suscep- 

 tible of popular explanation ; in fact, the especial services which M. 

 Hansen has rendered to astronomy consist in the development of new 

 formulae, and the exhibition of new artifices of calculation, in the 

 remotest and most abstruse departments of mathematical analysis. 

 Nevertheless, I trust I shall be able to convey such an idea of their 

 nature and object as will enable you at least to appreciate the mo- 

 tives which have influenced your Council in conferring on our il- 

 lustrious Associate this testimony of the Society's approbation. 



In proceeding to determine the motions of a celestial body urged 

 by a central force, and disturbed by the action of other bodies, the 

 accelerating forces in the direction of rectangular coordinates are 

 expressed by three differential equations of the second order, which, 

 as is well known, can only be integrated by approximation. To 

 obtain approximate integrals, two methods have been principally fol- 

 lowed. The first consists in deducing from the differential equa- 

 tions, expressions for the variations of the radius vector, longitude, 

 and latitude of the disturbed body in a function of the disturbing 

 force and its partial differentials ; and in integrating these expres- 

 sions, either by developing them in series which proceed according 

 to the powers of the eccentricities and inclinations, or else by the 

 method of parabolic quadratures. This is the most obvious method 

 of determining the perturbations, and also the simplest when the 

 approximations are only carried to terms of the order of the eccen- 

 tricities and inclinations ; but when a closer approximation becomes 

 necessary, and terms of a higher order are required to be included, 

 the expressions become complicated, and the method accordingly 

 loses its advantages. 



The other method of obtaining approximate results is known in 

 analysis as the method of variation of arbitrary constants. This 

 method, though undoubtedly entitled to be regarded as one of the 

 most ingenious artifices of modern analysis, is suggested in a man- 

 ner by the peculiar constitution of our solar system, in which the 

 disturbing forces which act upon any body bear so small a propor- 

 tion to the principal force which determines the general orbit, that 

 the body may be regarded as moving always in an ellipse, but in an 

 ellipse whose elements are in a state of continual though extremely 

 slow change. In accordance with this idea, the origin of which may 



