426 D'ALEMBERT. 



fashionable circles of Paris, there were not wanting those 

 who insisted that the whole fame of this great inquiry 

 belonged to Olairaut ; and it is painful to reflect on the 

 needless uneasiness which such insinuations gave to 

 D'Alembert. We shall recur to the subject after- 

 wards, and now must continue the history of this 

 problem. 



Thus, in investigating this famous "Problem of the 

 Three Bodies," all the three geometricians, without com- 

 municating together, took the same general course in the 

 field, like three navigators of consummate skill and most 

 practised experience tracing the pathless ocean, unseen 

 by one another, and each trusting to his seamanship, 

 his astronomical observations, and his time-keeper, and 

 all of them steering separately the same course. They 

 were each led to three equations, which nearly resembled 

 those obtained by the other two. Of the three equations 

 the most important is 



- 



a u dv 



-T-, I + u + - m- - = 



dV 



in which u is the reciprocal of the projection on the plane 

 of the ecliptic of the moon's distance from the earth, v 

 the moon's longitude with respect to the centre of 

 gravity of the earth and moon, P and T the resultants 

 respectively of all the forces acting on the moon parallel 



and perpendicular to -, and parallel to the plane of 



u 



the ecliptic, h an arbitrary constant. P and T being 

 complicated functions of the longitudes of the sun and 

 moon, as well as of the eccentricities of their orbits 



