^63 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



rJuLY, 



OF TERRESTRIAL OR HORIZONTAL REFRACTION, 



By Oliver Byrne, MHlheinatician. 

 Rays of light passing from objects whetlier terrestrial or eel 



tial, 



proceed in curves concave to the earth, thus: — rays of light (lassing 

 from objects T, S, to the eye of an observer at O, take curvelinear 

 directions T o O, S A O, instead of the straight lines T O, S O. 



Fig. 1. 



Now as the eye always traces the places of objects to the direction in 

 which the rays enter tlie eye, hence the observed elevations of objects 

 are always greater than the true one, for the direction in which rays 

 enter the eye is the direction of a tangent to the curve at the eye of 

 the observer. The objects T, S, will appear in the directions of the 

 tangents O T', O S'. The reason that the rays of light are bent in 

 passing through the atmosphere or part of it, is because the air is 

 densest at the surface and continues to decrease in density to the top 

 of the atmosphere : it is well known that a ray of light becomes bent 

 towards the perpendicular in being transmitted from a rare to a dense 

 medium, therefore the rays passing through the atmosphere are being 

 continually bent as they pass through a medium continually increasing 

 in density. The increase in the angular altitude of objects by being 

 observed in our atmosphere is called refraction : there arc two sorts 

 of refractions, horizontal or terrestrial, and astronomical. Horizontal 

 refraction affects objects situated in the atmosphere, astronomical re- 

 fraction is that which affects the altitudes of the heavenly bodies. 

 Refraction changes with the atmosphere, with regard to heat, cold, 

 humidity, dryness, &c.; when tables of refraction are given they are 

 calculated for a mean state of the atmosphere, in such a manner that 

 they can be made to answer any other state of the atmosphere with 

 some trifling allowance. 



The exact amount of terrestrial refraction is by no means satisfac- 

 torily settled : however it may be determined by the following me- 

 thod, for all practical purposes. 



Let A and A' be two elevated stations on the surface of the earth ; 

 B D the intercepted arc of the earth's surface ; C the earth's centre ; 

 A H' and A' H horizontal lines at A, A', produced to meet the oppo- 

 ste vertical lines C H', C H. 



Fig. 2. 





Let a, a', represent the apparent of the objects A, A', when viewed 

 from A' and A ; then is the angle A A' a, the refraction of A, and the 

 angle a A' a, the amount of refraction of A'; half the sum of these 

 two angles will be the horizontal or terrestrial refraction, supposing 

 it equal at each station. 



Now an instrument being placed at each of the stations A, A', the 

 reciprocal observations are to be made at the same instant, which is 

 determined by means of signals, or watches previously regulated for 

 that purpose : that is the observer at A, takes the apparent depres- 

 sions at A', at the same time exactly, that the other observer at A' 

 takes the apparent depression of A. 



In the quadrilateral A C A' I, the two angles at A and A' are right 

 angles, and therefore the angles at I and C are together equal to two 

 right angles : but the three angles of the triangle A A' I are together 

 equal to two right angles, consequently the angle at C which is mea- 

 sured by the arc B D is equal to the angles I A A' and I A' A taken 

 together. If therefore the sum of the two depressions H A' a and 

 H' A a' be taken from the sum of the angles I A' A, I A A' which are 

 together equal to C, (the angle C is known because its measure is 

 known ;) the remainder is the sum of both refractions, or angles A A' a, 

 A' A a'. 



Hence the following rule. 



Take the sum of the two depressions from the measure of the in- 

 tercepted terrestrial arc, and half the remainder is the fraction. 



If by reason of the minuteness of the contained arc B D, fig. 3, one 

 of the objects, instead of being depressed, appears elevated; suppose 

 A' to appear at a" above the horizontal line A H'. 



Then 



a A A' 4- a A' A = H A' a + H' A A' -1- o" A H ; 



= H A' A + H' A A' + a" A H' + H A a i 

 = C + a" AH'-HA'a: 



For C = H A' A + H' A A'. In this case, because a" A A' = 

 a A' A, we have this rule .- — 



To the contained are add the elevation, from this sum subtract the 

 depression, and half the remainder will be the refraction. 



As we have previously remarked, the amount of terrestrial refrac- 

 tion is found to vary considerably with the different states of the 

 atmosphere ; it is stated in the account of the trigonometrical survey 

 of England (vol. I. p. 160 — ■dbii) that, the quantity of terrestrial re- 

 fraction varies from } to —^ of the contained arc. 



Although every practical man considers the amount of terrestrial 

 refraction to be more or less according to his experience, yet all range 

 between f and -Jj of the contained arc. Dr. Maskelyne considered it 

 to be ^, M. Legeudre ^, M. Delambre y^, Mudge and his com- 

 panions Y5, at a medium. A similar table to the succeeding, would 

 be convenient to engineers, when they have made up their minds 

 which of the fractional parts of the contained arc, that range between 

 i and Ys, best suit their purpose. 



Examples. 

 I. Suppose the angle of depression of an object five miles distant 

 from the place of observation, to be 3° 47' 45", what is the true de- 

 pression, supposing we take Dr. Maskelyne's allowance?, 

 5 miles := 26400 feet, 

 fij of 26400 = 2640 

 20" — 2028-6 



6" 



61 1-4 

 ■ 608-59 



