120 Mr DAVIES on the Nature of the Hour-Lines 



different from any that were entertained by the Egyptian, Babylonian, 

 or Grecian philosophers, that we have discovered the true character of 

 the lines ; and it has been mentioned by Mr CADELL, and calculated by 

 DELAMBRE, how little these lines in the intertropical regions deviate from 

 their chords, in right ascension. This, however, is more distinctly brought 

 under our notice by the fact, that the developed hectemoria is identical with 

 a series of harmonic curves, the characters of which have been often examined, 

 and are well understood. It is not easy, nevertheless, to ascertain by direct 

 investigation the greatest error that can result from the adoption of the 

 hectemorial chord for the curve itself, between certain assigned limits of de- 

 clination D'. Treated by the usual methods of maxima and minima we are 

 led to an equation of the form 



90 



E. sin n L = cos- 



n 



This equation, it is at first sight obvious, can be solved only by a tedi- 

 ous approximation, the convergency of which is not only slow, but the direc- 

 tion and amount of the correction at each step, very difficult to assign. DE- 

 LAMBRE, therefore, calculated a few of the errors which belonged to diffe- 

 rent declinations and latitudes, and for each value of n which entered into 

 the expression for the twelve hour-lines. These he found to be very mi- 

 nute. 



It is still possible, by means of the direct calculus, to find a quantity for 

 each hour-line in a dial adapted to any latitude which shall differ but little 

 from the maximum error, and that quantity at the same time shall be al- 

 ways greater than the maximum error. This is founded on the same pro- 

 perty that the developed hectemoria becomes the Taylorean Curve. 



Let A be the origin, AB the axis of the 

 harmonic curve, find the angle B which corre- 

 sponds to the value D' of the declination and 

 for the hour-line in question. Let C be the 

 point in the curve for which the error in right 



