140 Mr. S. Sharpe on the Theory of Differences. 



Hence, with seven observations, 

 ?^j = mean of observations —2 mean of second differences, 

 and thus we may construct the following table : 



Remai'k, — This proposition may be used on all occasions 

 to which the rule is applicable, that " by a single observation 

 is meant the mean of a series," (see Requisite Tables, Baily's 

 Tables, &c.) and in particular in learning the time from a 

 single altitude observed ; when we consider that many persons 

 have occasion to learn the time, who have neither instrument 

 to observe a transit, nor leisure to take equal altitudes. 



Prop. IV. — From an observed place of the moon, and four 

 given places in the Nautical Almanac, to determine the time 

 at Greenwich. 



This is I believe not in general solved analytically, but syn- 

 thetically; thus, 



1st, Assuming the time at Greenwich, by help of the differ- 

 ence of longitude known approximately. 



2ndly, By the theory of differences determining the moon's 

 place by equation [1]. 





h'{h'-h) h' ih'-h)ih'-2h) 



.Srdly, Thence correctly the assumed time. 



This equation is computed more conveniently by means of 

 tables of the coefficients of A, A^ and A^j but if such tables are 

 not at hand would be more easily computed in the form [2]. 



h' / , A'?< ^^7i\ /i'-/A-u A^u\ /i"/A'u\ 



^"=t-1^"--2-+-^;+M"2--^j+f(-6-j- 



But 



