Dr. Hargreave on the Problem of Three Bodies. 471 



disturbing force, or rather of the ratio of the disturbing force to the 

 central force ; but that it may remain precisely the same, though this 

 ratio should be diminished or increased without limit. The differ- 

 ence affects not the aggregate amount of de\dation or disturbance 

 caused, but the time in which this aggregate amount is produced. 

 If we consider the undisturbed problem of two planets about a suu 

 as representing motion in two planes inclined to each other at the 

 angle I, and in ellipses having eccentricities <?, and ^3, it is shown 

 that, no matter how small or how large may be the disturbing force 

 produced on each orbit by the other planet, the aggregate amount of 

 disturbance of the planet m., is of tlie order of the quantities I and e„ 

 and that of the planet m^, of the order of I and e.^. From considera- 

 tions of this nature, which are dwelt upon at length in the memoir, 

 the author concludes that the ordinary direct methods of solution 

 by approximation, being based upon the erroneous assumption that 

 the variations of the coordinates are of the order of the disturbing 

 force, are not, in a mathematical sense, legitimate processes ; and 

 that, in the planetary theory, they produce results practically true 

 only on account of the minuteness of the disturbing forces, and the 

 consequent great length of the secular periods ; and that, in the 

 lunar theory, their failure is made evident, in consequence of the 

 comparatively large magnitude of the disturbing force, and the con- 

 sequent rapidity with which the elements of the moon's orbit pass 

 through their secular periods. 



The formulae for the variations of the elements are then applied 

 to the lunar theory ; and some of the integrations are effected 

 by means of a lemma containing the solution of the differential 

 equation 



d(p ^ . 

 -^=F cos (pt-qf) 



(where F, p and q are numerical coefficients), in the form 



- qF+pcosMt ^, ^ . ^, q-F-{ 

 cos ipt-i<P)=j,^^Y cosMt' ^^ ^""S (^—y-)P- 



By this method, the total motion of the moon's perigee, as well as 

 the coefficients of the evection, are fully obtained in the first in- 

 stance, without the necessity of Jiny second aj)proximation ; and the 

 usual difficulty as to the movement of the perigee does not present 

 itself. Tiie motion of the node, and tlie evection in latitude, are 

 correctly obtained in a similar manner. 



This jiart of the memoir is concluded by an extension of the 

 general formulie for the tangential variation of elements to the case 

 in which we suppose the constant f.i to become variable, the result 

 being to add to each variation a term involving Cfx. 



The tliird part of the Pajjcr contains the development of the 

 method of osculating variation, before briefly described ; from which 

 are deduced the formula: for the osculating variations of elliptic ele- 

 ments. This method is capable of being apj)lied to the planetary 

 Bud lunar theories, &i well as that of tangential variatiou ; but the 



