Mr. S. Sharpe oh the Theory of Differences. 139 



Prop. 11. — To determine the point of contrary flexure in a 

 curve ; or to interpolate a term in a series, such that its second 

 difference is = by means of the theory of differences. 



Instead of ?/^ + ?/,, as in the former Proposition, we now have 



Mi^v _ y/^ Consequently by expanding the three equations 

 [3] [4] and [5] we have 



0=^^ + -^^^ + h^ ^• 



Dividing by 4^, and rejecting the terms then containing a, 



we have o = B + -^3C + -^6D 



/i' _ ^ , 6B^D 



~ IT ~ 3 C "^ 27 O 

 and hence we have in equation [l] the value of «'. 



By this proposition and a series of altitudes of a star in the 

 east or west, we may deduce the time of its passing the prime 

 vertical, and its altitude when there. 



Hence if P be the hour-angle at the pole, 

 L, the latitude of the place, 

 A, the altitude on the prime vertical, 

 D, the star's declination, 



sin P = -^ and sin L = 4^ (See Baily's Tables.^ 

 cos \J sin ii 



Pro-p. III. — From a set of observations made at equal in- 

 tervals to obtain one more correct, corresponding to the mean 

 of the times. Or it may be stated, To correct the mean ot a 

 set of observations by help of the theory of differences. 



Let there be seven observations ; then 



u — u 



u^ =^ u + A « 

 u^ = u + 2AM + A^« 

 »/3 = M +3AU + 3A^M + A« 

 u. = u +4>Au + 6A'm + *A'« 

 u. = u + 5AU + lOA^u + IOA'm 

 Mc = « +6AK + ISA'u + 20A='m 

 neglecting the higher differences. 



Now the mean of these /juantities gives us 



7/., = u + 3 Aw + 5A'^M + 5A^«, 

 which is evidently too great by 2 A'" + * A'«- 



1' 2 Hence 



