330 Secular Acceleration of the Moori's Mean Motion. 



Substituting e^ in place of {x^ -{-y"^) in the foregoing expres- 



1 3e2 

 sion for the mean value, it becomes ~+— y. This expression 



contains only constant quantities, and since ED and HN are any 

 conjugate diameters, it is true for the whole orbit. Therefore 

 the first term in our proportion, which we wished to find, viz. 

 the mean of the reciprocals of the cubes of the radii vectores of 



1 3e3 



an ellipse, is —,+-^' 



The mean distance of the earth from the sun is equal to the 

 semi-transverse axis of its orbit, and the second term in our pro- 



1 



portion is therefore — • Multiplying both these terms by a^ to 



render the proportion more simple, and stating it in form, it reads 



3e2 

 thus, 1+ — - \ 1: :the actual diminution of the moon's gravity : 



the diminution if the earth revolved in a circle at the same mean 

 distance, which we have shown to be ^l_ of the earth's attrac- 

 tion. In this proportion the second and fourth terms are con- 

 stant ; therefore if the first varies, the third must vary in the 

 same ratio. Now we remarked near the commencement of the 

 article, that e, the eccentricity of the earth's orbit, is diminishing 

 about 4 oirs every year, and as the squares of numbers where 

 the difference is small vary about twice as much as the numbers 

 themselves, e- diminishes about 4 oItj ^^^ ^ year. Further, e is 



3e2 

 about gV of a, and consequently — ^ is about y 3'^ „ of 1 ; there- 

 fore the annual diminution of the first term in the proportion is 

 4oIt5 of Tjoo of the second, and the third must diminish by the 

 same fraction of the fourth. But the fourth term is ^\j of the 

 moon's gravity toward the earth. Hence the annual diminution 

 in the third term is j^fy^ of y^Vo of sir^sTs 2^29 of the 

 moon's gravity toward the earth. That is to say, the sun dimin- 

 ishes the gravity of the moon toward the earth less and less eve- 

 ry year by the value of this fraction, which therefore expresses 

 the annual increase of gravity. 



This variation in gravity affects the moon's periodic time in 

 two ways ; 1st, by contracting its orbit so that it has a less dis- 

 tance to travel ; and 2d, by increasing its absolute velocity. 

 The first is obvious, and to show its amount let g represent the 

 moon's gravity toward the earth on any given year, and r the 



