138 



and 



Mr. S. Sharpe on the Tlieory of Differences. 



'h'-a> 



by expanding the binomials and subtracting one equation from 

 the other, we have 



o = -T- +-;?- + — V — ^+ h'^ ^ 



2a 

 then dividing by— y— , and, as a may be infinitely small, re- 

 jecting those terms which then involve o, 



= A+-^2B+-^3C + -^^4D 

 whence by Newton's method of reversing a series, we have 



A' 

 when ti' is a maximum ; and with this value of -7— we shall 



n 



know from our original equation the value of li as a maximum. 



As an example I shall take the following declinations of 



the sun from the Nautical Almanac for 1829. 



in which A = +54"-4, B = -11", C = -0"-38. 



D being + 0"*05 may be neglected. 



A' 



' — 2-46 — 0-31 + 0"-24, and h' = 2 days 9" 15™, 

 h 



being three hours wrong ; and 



u' = 130" - 62"'8 - 4"-9 + u = 23° 27' 33"-3, 

 being 0"*5 wrong. 



These results are as accurate as could be expected, consi- 

 dering the original values of u do not contain tenths of a se- 

 cond ; and if the value of -j- had been known from other 



data, the resulting value of u' would have been exact. 



Prop. 



