Secular Acceleration of the Moon's Mean Motion. 327 



The proportion for the other force into which m was resolved, 



viz. in the direction ES, will read SM : SE: :wi : the force re- 



. , SE 

 quired = g^w. But the earth is attracted by the sun in the same 



direction ES, and it is the difference of the attractions only that 

 exerts any disturbing influence on the moon in this direction. 

 We will therefore find how much the attraction of the sun or 

 the earth is, and subtract one from the other. By the laws of 

 gravity SE^ : SM^ : :m : the sun's attraction at the distance SE. 



SM^' 

 Hence the earth is attracted with a force equal to opjwi. Re- 

 ducing the fractions to a common denominator and subtracting 



SE^-SM^ 

 we have gp2 xSIVT ^' ^'^^ SM=SE — EF very nearly; and 



by involving both sides of this eqimtion, rejecting the third and 

 fourth terms in the right hand member on account of their small- 

 ness, and transposing, we have SE^' — SM^'^SSE^ xEE. Sub- 

 stituting this value in place of the numerator of the above frac- 



SSE^'xEF 3EF 



tion, it becomes opa vSM ^' which is equal to -Qirrw, and is 



the disturbing force in the direction ES. Resolving this force 

 into two others, one in the direction EM and the other at right 

 angles to it, i. e. in the directions EG and GS, the proportion for 

 the former or ablatitious force will read SE : EG or (since the 



opp 



triangles EGS and EFM are similar) ME : EF::-g^w : the 



3EF2 



ablatitious force = g]yr v. me ^' 



To find the mean ablatitious force for the whole of the moon's 



orbit, take three other points M', M'^ and M'^' at the same distance 



from B and C that M is from D. Connect these points with E, 



and from them draw WF', M^F'^and M'^'F''', perpendicular to CD. 



According to the expression last found, the ablatitious force for 



3EF2 3EF2' 



the four pomts M, M', M'^ and M'^', is gj^^^j^^ m, gj^/^j ^^, 



SEF^" 3EF^''' 



SM^ME''* ^"^ SM^llE^' ^^ ^'^^ ^^® "'^^^ ^^^ "^^'^' ^^^ 

 them together, and divide by 4 ; or since the denominators are 

 nearly equal, we may without much error take the mean of the 

 numerators and of the denominators as the mean value of the 



