215 



Suppose, now, we make this reduction, and collect the results, as 

 given by equations (A^, (i^), and ((7), and we have, finally, 



du> 3 ^ 135 / jll x , 3 B- A 1 1 ( 1 1 \ 1 



It ~ "32"V ?T / ~D n - n l \n l + 6 . 2n - Su 1 J ¥ 



2 (m(3(^> + yfa) - * 3 )) 2m 



It contains, therefore, a constant term ; and hence a z will contain anew 

 periodic term multiplied by the time. 



Corresponding also to equation (A), there will be a term arising 

 from the second term in the first part of JV, viz., sin 2 * sin2<p, and in 

 (B) a term arising from cos<p in the equation for « 1} &c, as has been 

 before mentioned : but these destroy each other. 



There will also be constant terms depending upon the arguments 



6\ 0 - 26", and 20 - 0\ 



which arise in the same way as those given above ; that is, in every 

 case the term arising from a x a. 2 on the left side is identically de- 

 stroyed by a term on the other side, while on the other hand the 

 first term of the function iV^will produce, just as above, a term which 

 is not destroyed, and another arising from the substitution of the 

 next approximate values of <w n &c, in the second part of -N\ also in 

 all cases the constant term produced by the function sin H sin 2<p in the 

 first part of N is identically destroyed. The constant terms, however, 

 in all these latter cases, are multiplied by sin 2 *, and depend upon quite 

 different constant qualities. The one which has been examined ap- 

 pears to be the most important, and is the only one which would exist 

 independently of sin i ; that is, when the plane of the equator does not 

 differ sensibly from that of the orbit. The rest it is needless to say 

 more about at present ; and it only remains to discuss that already 

 found. 



Discussion of the foregoing Results. 

 (1.) Application to the Earth. 



The principal object there is to ascertain whether sufficient change 

 has been produced in the motion of the earth to be perceptible by the 



observations of the last 2000 years. In the term -y which occurs, put 



r - a, and multiply numerator and denominator by fi + fi x where ^ is 

 the mass of the earth, it becomes 



— — , or, putting — — = n'\ 



fx + /*! a? x a 3 



we have / /u, y / ^ y , 6 



And for the moon, disturbed by the earth, we should have for the cor- 

 responding quantity 



