in Laplace's Theory of the Tides. 239 



convergence to the true values of R^-j, Rfc-aj & c « If we take 

 llft+i exactly equal to =-^ — -, we gat R/ c = cOj R yfc _ 1 = ? an J 

 then rapid convergence to the true values for Ri_ 2 , &c. 

 But if we take for R i+l a value less than -^ — - by a certain 



very small difference, we find for R-. a value less than — =- • 



: . . 2& + 4 



by a corresponding very small difference, and then for Uk-i a value 



2# i 



less than — — ^-bya corresponding small difference, and so on. 



Any value of Rfc +1 , except precisely the one particular value last 

 indicated, will, provided k be large enough, lead to the desired 

 values of the lower ratios after the two, three, four or more suc- 

 cessive applications of the formula required to dissipate the 

 effects of the initial error. It is a curious and instructive arith- 

 metical exercise to calculate R k , T&k-i, an( l s0 on. down to R x ; 

 and then by successive reverse applications of the formula to cal- 

 culate R 2 , R 3 , . . . R&-1, Rjfc. If the calculation has been rigo- 

 rous, of course the initial value of R^ will be that found at the end 

 of the process; but if the calculation has been approximate (say, 

 with always the same number of significant figures retained in 

 each step), the value found for R& will be not the initial value, 

 . , 2& + 1 . 2k + l e 



but 5+4- or > more a pp r ™ tel y> 2 TTi-(^i)(Fr2) e 



And if we choose for Rj any other value than precisely that ob- 

 tained by an infinitely accurate application of Laplace's process, 

 then work up by successive reverse applications of the formula, 



2/fc-f-l 

 we find for R^. a value approximately equal to ,— r — -*. Laplace 



* Compare with the calculation of the formula 



1 



2a- n- 



where a denotes any numerical quantity > 1 . Take any value at random 

 for r , and calculate r lt r 2 , r 3 . . , by successive applications of the formula. 

 For larger and larger values of i, r* will be found more and more nearly 

 equal to the smaller root of the equation 



x 1 -2ax-\-\=Q. 



Now calculate backwards to r by the reversed formula 



n 



and instead of finding the initial value of r again, the result (unless the 

 calculation has been rigorously accurate in every step) will be approxi- 



