30] FOR THE MOON'S PARALLAX. 257 



of the radius vector to the same order of accuracy as that which Delaunay 

 has already attained in the case of the corresponding expressions for the 

 longitude and latitude. The work would be one of simple substitution, 

 not requiring the solution of any new equations, and consequently its only 

 difficulty would consist in its great length. 



The fact that Delaunay's determination of the value of the reciprocal 

 of the radius vector is a comparatively rough one, affords a ready explanation 

 of a difficulty which Sir George Airy has recently met with in his Numerical 

 Lunar Theory. 



The first operation required in this method is the substitution in the 

 differential equations of motion of the numerical values of the Moon's 

 coordinates as obtained in Delaunay's theory. If the theory were exact, 

 the result of the substitution in each equation would be identically zero, 

 so that the coefficient of each separate term in the result of the substi- 

 tution would vanish. In consequence of errors in the coefficients obtained 

 by Delaunay, however, this mutual destruction of terms will not take place, 

 and the result of the substitution will consist of a number of terms the 

 coefficients of which will depend on the errors of the assumed coefficients. 



If, as is actually the case, these latter errors be so small that their 

 squares and products may be neglected, each of the residual coefficients 

 may be represented by a linear function of the errors of the assumed 

 coefficients, and the formation of the corresponding linear equations constitutes 

 the second operation in Sir George Airy's method. The solution of these 

 linear equations by successive approximations will finally give the corrections 

 which must be applied to Delaunay's coefficients in order to satisfy the 

 differential equations. 



Now, since the proportionate errors of Delaunay's coefficients of parallax 

 are considerable, and much greater than the errors affecting his coefficients 

 of longitude and latitude, it will be readily understood that the result of 

 the substitutions will be to leave considerable residual coefficients in the 

 two equations which relate to motion parallel to the ecliptic, and much 

 smaller residual coefficients in the third equation which relates to motion 

 normal to the ecliptic, since in this last equation every error in the co- 

 efficients of the radius vector or of its reciprocal will be multiplied by the 

 sine of the inclination of the Moon's orbit. This result, which might thus 

 have been anticipated, is exactly what Sir George Airy has found to take 

 place, according to a memorandum which he has recently addressed to the 

 Board of Visitors of the Royal Observatory. 



A. 33 



