Royal Astronomical Society. 525 



to an entirely new mode of conducting the numerical calculations ; 

 so that, if it cannot be said that he has furnished us with a new in- 

 strument wherewith to attack the difficulties of the problem, he is 

 at least entitled to the merit of having taught us a new method of 

 applying that of which we were already in possession. 



On taking a general view of Hansen's method*, the point which 

 first presents itself as remarkable, and that indeed in which the 

 novelty of his process essentially consists, is the original and highly 

 ingenious artifice which he employs in order to arrive at the ex- 

 pressions for the 'perturbed coordinates, — namely, the longitude, 

 radius vector, and latitude. In the usual method of proceeding, the 

 arbitrary constants introduced by integration are determinate func- 

 tions of the elliptic elements and time, and the perturbations of co- 

 ordinates are obtained by supposing the elements to vary. Instead 

 of the true time, M. Hansen introduces into the functions an ana- 

 logous, but indeterminate quantity, and considers the elements as 

 invariable. He then determines the variations which this quantity 

 must undergo (in other words, he finds what alteration must be 

 made in the time, in the place where it enters explicitly into the 

 elliptic formulae), in order that the elliptic formulae, with altered 

 time and invariable elements, may give the same value of the inde- 

 terminate functions as would be found by using the true time and 

 variable elements. Suppose, for example, the function of elements 

 and time to be the true longitude ; then the problem, according to 

 M. Hansen's method of viewing it, amounts to this : — To find the 

 perturbations which must be applied to the mean longitude, in order 

 that the true longitude deduced from it with the use of invariable 

 elements, may be the true perturbed longitude. 



It is evident, that the use of invariable elements, and time altered 

 so as to give the correct value for longitude, would not, with the 

 elliptic formulae, give a correct value of the radius vector ; but this 

 difficulty is surmounted in an extremely ingenious manner by the 

 introduction of subsidiary terms, which, being applied as corrections 

 to the radius vector of the unaltered elliptic orbit {i. e. unaltered 

 except in time), give its true perturbed value. By similar considera- 

 tions an expression is found for the latitude in the disturbed orbit. 

 It would be impossible, however, without the aid of algebraic sym- 

 bols, to give an idea of the analytical processes employed for deter- 

 mining these subsidiary terms ; and for the same reason I must con- 

 tent myself with a bare allusion to the still more remarkable artifice 

 to which he has recourse in order to obtain an expression for the 

 continuous variation of the perigee and node of the lunar orbit, for 

 which, by reason of their rapid revolution, invariable elements will 

 clearly not suffice, and a departure in some degree from the original 

 principles becomes necessary. 



These deviations from the usual methods lead to very important 

 advantages in the calculation of the tables, for the series expressing 

 the perturbations of coordinates are not only rendered more conver- 



* [On the subject of M. Hansen's method see Phil. Mag., Third Series, 

 vol. xix. p. 82.— Edit.] 



