414 D'ALEMBERT. 



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 



dv 



T 



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 result- 

 ants respectively of all the forces acting on the moon 



parallel and perpendicular to , and parallel to the 



plane of the ecliptic, h an arbitrary constant. P and 

 r being complicated functions of the longitudes of the 

 sun and moon, as well as of the eccentricities of their 

 orbits, have to be developed for the further solution of 

 the problem. 



Now, it is a truly remarkable circumstance that the 

 conclusion at which all these great men separately 

 arrived was afterwards found to be erroneous. They 

 made the revolving motion of the moon's apogee (or 

 the revolution which the most distant part of her orbit 

 makes in a certain time) half as much as the observa- 

 tions show it to be ; and in a revolution of the moon, 

 1 30' 43", instead of 3 2' 32" the observations giving 

 about nine years for the period, which the revolution 

 really takes, instead of eighteen. Clairaut first stated 

 this apparent failure of the Newtonian theory, and as 

 he had taken pains to make the investigation " avec 

 toute 1'exactitude qu'elle demandoit," ('Mem.' 1745, p. 

 336,) he was with great reluctance driven to conclude 



