25] RECIPROCAL OF THE MOON'S RADIUS VECTOR. 193 



Thus we find 



d"y l d"y\ I d\ d* 



+ /* (a^i + yyi + zzi) -7 - 4) -0; 



\'J ' I 



or 



d I dx, dx\ d I dy, dy\ d I dz l dz 



I ry* _ _ rf _ / nt ' 1 _ n, _ ^_ II - - [ *? __ _ _ ? _ 



dt dt ' dt) + dt \ y dt Jl dt) + dt \ dt ' dt 



+ p- (xxi + yyi + zz,} (, --f)-0. 

 \'i ' / 



Hence the quantity 



is a complete differential coefficient with respect to t, and therefore when 

 developed in cosines of angles which increase proportionally to the time it 

 cannot contain any constant term"". 



Now 



and 



\?y r-; \1\ rj \j-i\ \i\ 77 j 



Hence, if x x lt y y lt z z 1( and therefore also r )\ , and ; be quantities 

 of the first order with respect to any symbol, then 



1 1 



will differ from 3 ( --- ) by a quantity of the third order only. 

 \r, r] y 



* We may remark here that neither of the quantities 



JL_1 



can contain any constant term, but no use is made of this in what follows. 

 A. 



25 



