526 Prof. Challis on the Theory of the Moon's Motion. 

 Putting now the equation (C) under the form 



»'V' 



y^_Cr«-A»+2/tr+^ 



it follows from what is shown above, that the integration must 

 be performed as if the quantity under the radical were of four 

 dimensions with respect to r, the last term being treated as vari- 

 able. This cannot be done exactly, but an integral sufficiently 

 approximate for our purpose may be obtained on the same prin- 

 ciple as that applied to the approximate solution of algebraic 

 equations of high dimensions. That is, having ascertained in 



the manner exhibited above that '- \% an approximate value of r, 



we may substitute in the above equation ^ -f ?; for r, and expand 



to the second power of r, to secure an approximation of the first 

 order. This being done, the equation is integrable, and the 

 same results are obtained as those given in mv communication 

 to the Philosophical Magazine for April 1854 (p. 281). It will 

 only be necessary to insert here those results which may be useful 

 in the second approximation. 



c(«*+.+y)=cos-.^-(.«-(l-9')* 

 - =1 + 6 cos c(^4-y) 



m being the ratio of the moon's periodic time to the sun's, and 

 a and e being new constants, the relations of which to the con- 

 stants h and C are given by the equations 



From the last two equations may be deduced the following : 

 Hence 



A: 



(■4) 



terms higher in value than the last term, and the constants C and h must 

 he related to each other in such a manner as to satisfy this condition. 



The relation between C and h thus obtained is the same that results by 

 supposing the expression to contain the disturbing force as % factor. 



