[ 13V ] 



XVII. On the TJieory of Differences, by Samuel Sharpe, 

 Esq. F.G.S.* 



Prop. I. — nnO interpolate a maximum or minimum in a 

 -*• series, by means of the theory of differences. 



This perhaps may be useful on many occasions, in practical 

 astronomy, when the observer has a series of altitudes at equal 

 intervals, without all the data, such as time and latitude, ne- 

 cessary for the more rigid formula?. As for instance ; From 

 a series of altitudes near the meridian, to find the meridional 

 altitude and approximately the time of transit ; which is a far 

 more accurate method than that commonly used by sailors, 

 of watching the sun's ascent till it becomes stationary, which 

 gives the time of noon with extreme inaccuracy, and the me- 

 ridional altitude single and uncorrected by others. 



And again ; From a series of declinations near the solstice 

 to determine the solstitial declination. 



To explain the signs make 



and m' and /i' the quantities sought. 



By a well-known formula in the theory of differences, we 

 have 



h' h'<Ji'—h) ,, h'(h'—k)(h'—2/iA^ . 



h 2h- b h^ 



and by expanding and rearranging, [l] 



which for convenience we will write 



and if we consider this value of u as a maximum, we may 

 make two equal values, one on each side of it; vi/. 



• Communicated by the Author. 

 N.S. Vol. 10. No. 56. An^. 1831. 



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