52 DYNAMICAL PRINCIPLE. 



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 



^ + + T^r- ~ = o, 



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 the ecliptic, h an arbitrary con- 

 stant. P and T 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 

 observations 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 IS ewtonian theory, and as he had taken pains to make 

 the investigation " avec toute 1'exactituclc qu'elle demandoit " 



