Dr. Arnott on the Measurements of Heights. 1 7 



the first, D, T, H, iu the second, and S, r, », in the third, be accirratcly 

 examined, we may (by simple equations) find z = 



_1_ yy,dl> (T— — S (dR (r —t) + J) h (T — r)) _^ 

 d S D V (d — D) (r — -^{d — S) (T — t) 



1 / 5 (T — t) ( h J) — Rd) — T) jT — t) ( hi — y, d) , 

 dlB ^ id — D) (r — - (c? — 5) (T — " ' 



sdD — (AD — He?) -, h — d(vt+2d) 



y- — dDiT^r-^"^-"=-; — -d ^ 



which values being substituted in the equation d {x + y t + 2d) gives 

 us a formula for that thermometer, and the decrement of atmospheric 

 temperature for the climate, whatever it be, so that we may calculate by 

 it other elevations. 



I shall now illustrate some of the formulas given above by five 

 examples 



Ex. 1 . Dr. T. Thomson writes that at one place, which he considered 

 to be at an elevation of about 18,.500 feet, he observed the boiling point 

 to be 1 80"-3 Fahr., which corresponds to 82''-39 Cent. Now I am ignorant 

 under what pressure his instrument is graduated, and also of the temper- 

 ature of the external air. The former probably showed 100° when the 

 barometer stood at 30 inches, and the latter, from the season of the year 

 and collateral remarks made by Dr. T. Thomson, may be assumed to be 

 freezing, or 0°. In formula S, then, t ^ o,~B — 6 = 17'6], whence the 

 height = 35'22 (490 + 35-22) = 18,498 feet, difi"ering only 2 feet from 

 the height suggested by Dr. Thomson. But if we take the foraiula U, 

 which I prefer, the height is 18,674, or 172 feet more than supposed by 

 Dr. Thomson. 



Ex. 2. In the extract from Dr. T. Thomson's letter, at the commence- 

 ment of these remarks, he states that his thermometer showed the boiling 

 point to be 203°'6, while the temperatiu-e of the atmosphere was 80°. 

 These on the Centigrade scale (which I prefer for its simplicity in calcu- 

 lation) are 95°-333, and 26"-667 respectively : hence t — 26°-677, and 

 B — h = (supposing B =: 100°, as in the first example) = 4°-667. 

 The height, by formula S, is thus 5156, or by formula U, 5203 feet. By 

 using Fahrenheit's thermometer, the formula X gives the height = 5147, 

 and W, 5198. Unfortunately, Dr. Thomson does not state what he 

 considers the height of this station to be. 



Ex. 3. Humboldt observed the barometer on Chimborazo to be at 

 0-377275 metres, or 14-853 English inches; and the detached thermo- 

 meter to indicate — l"-6 Cent. The boiling point of a thermometer 

 adjusted to 100" Cent, at a pressure of 30 inches, would thus, by my 

 formula (60-6 Log. 2^ -f 10-486 = h) be on the mountain 81°-5. In 

 formula U, then, t = — l°-6, B — b ^ 18-5, whence the height = 

 37 (495 —3-2 -I- 37) = 19,566, The true height, after making every 

 correction for the hygrometer, latitude, &c. &c., is 19,441 : the error, 125 

 feet, arising principally from it requiring 700 feet of ascent for each 

 degree, whereas I have only allowed 500. 



Vol. III.— No. 1. 2 



