Heights of the principal Hills of Dent, fyc. Yorkshire. 195 



tact with the ground at the station is rarer in the heat of the day 

 than the stratum immediately above it, and denser in the morn- 

 ing and evening, than comports with the natural constitution 

 of the atmosphere. The refraction varies in consequence from 



~ + x in the morning to x, its amount about noon, 



and reverts in the evening to the first expression of its value. 

 Hence the refraction on similar days is perpetually changing, 



and becomes a constant ratio of the arc (—} only twice in 



the day ; viz. in the middle of the forenoon and afternoon, when 

 x or the maximum increment or decrement of the regular re- 

 fraction resulting from the passage of the ray through the dis- 

 turbed stratum of air, becomes 0. In such a condition of the 

 atmosphere it would evidently be impossible to ascertain the 

 correct angle for calculation from even contemporary reciprocal 

 observations, unless x and its horary variations should have 

 the same value at the same time at both stations. 



On a careful consideration of the errors to which the determi- 

 nation of the angular difference of level is liable, we cannot but 

 admit that the least objectionable method of measuring the 

 difference of altitude of two hills is to make the requisite ob- 

 servations at an intermediate station. When the latter is equi- 

 distant from the former, so complete a compensation of errors 

 (that arising from local attraction excepted) is effected, that it 

 becomes superfluous to be acquainted with the error of colli- 

 mation (from whatever cause arising), or even the refraction, 

 although variable to a small amount, provided the two obser- 

 vations were nearly contemporary. But if we suppose the two 

 hills AB, and the intermediate station S, to be in a line and 

 equal in altitude, and admit the plumb-line to be drawn 10" 

 towards A, then shall we make A lower, and B higher than S 



by about — * 1S ncc , and B will be calculated to be higher 



than A by twice this quantity. As it may frequently be im- 

 practicable to have an equidistant station, we must select two 

 (or more) places ST, so situated that the distances S A + TA 

 may be nearly equal to SB + TB. In this latter case the error 

 of collimation and the refraction should certainly be known 

 approximatively; but great errors may be committed in this re- 

 spect without materially affecting the difference of altitude of 

 the two hills. When SA is more nearly equal to SB than 

 TA is to TB, the difference of level is probably more cor- 

 rectly determined from S than from T, unless S A + SB should 

 greatly exceed TA + TB. Hence the following correction : 



2 C 2 Let 



