4

'The Resolution of some General Problems

" i— 0,0 5 50 5 ^^ 2 + 0,007 27 6 cos./>z— 0,0 u 27 cos. p— 0 .z|-u;=*X 4~°» 00020 4 co s. /> + g - 2 -{- 0 , 0010623 cos. /,- 2 /Z.L- 0,C00023 Cos. 2 pZ

-f 0,00Q044Cos. 2 fi-g.Z-{ - 0,00001 2cps. 20Z-fO, 00000700 s 2p-2;.Z. ^- 0 ,00002 I Cos. p - 312 .Z

But this value must now be corrected by the difference arisingfrom our having, in all the preceding calculations, taken the di-visor bb= 1, instead of the true value 1 -j-'|xBB-J-CC-PDD&cJ 1(vid. p. 159.) which value, because B=o,05505, C =—0,007276 &c. is given =1,0097. From whence and the equa-tions, on p. 153 and 163, it appears, that all the terms aboveexhibited, in whose exponents the quantity p \s,Ji?:gly, concerned,ought to be diminished in the ratio of 1 to 1,0097 ; ^nd thatall thos 3 where 2 p is in like manner concerned, ought to be di-minished, in she duplicate of that ratio: by which means ourequation is, at length, reduced to

f 1—0,05505 cos£z-f-o,c07206cos./>z—0.01116 cos. p —p.z]-xv = x ^ -fO, 000202 cos.^, 3 .z-f0,0010 5 2 c«s. p —2,3,2.-0,00002 2 cos. 2 p %I q- 0 ,C 00 G 43 C 0 s. 2 />- 3 .Z-(- 0 , 0 C 00 ] 2 cus.232-^0, 0 C 2007 Cvs. 2 p— 2 j 3 .a( -j- 0,00002 I cos. 7 - 33.2.

From whence all the great equations of the moon’s motion,and all the smaller ones, except those depending on the sun’sexcentricity &c. are obtained, within ieis than half a minuteof the truth; supposing the mean excentricity (B) to be heretruly assigned. If it should be found necessary to augment, ordiminish the value thereof, dien the term — 0,011 i6cofy> —fi.z(producing the equation , called the eve B ion) must be also aug-mented or diminished, in the same ratio; and the term-j-0,00 1 052 cos. p — 2/3 .z (which is the next considerable of thosewherein fi enters) must be augmented or diminished in theduplicate of that ratio. As to the rest of the terms, they areso small, that a little alteration in the value of B will produceno difference in them worth notice.

In the same manner, the inequalities caused in the moon’smotion by the fun’s excentricity, may be computed. For themean motion of the moon being given, very nearly, by the pre-ceding calculations, the mean motion of the fun, being in pro-portion thereto as m to 1, will be also known; from whence,and the excentricity , the sun’s true anomaly , and distance from

the