152 Royal Society : — Mr. Ellis on the Corrections for 



a f , a" the corresponding temperatures of the air for any three sta- 

 tions, then the formula (a) shows that, rejecting the small correc- 

 tions v v Y v the height, as determined by the total method, will be 

 the same as that determined by the gradual method when 

 (a+a!>). (log /3-log/3») = 



(« + «') • (log/3-log /3') + (a' + a") . (log /3'-log /3"), 

 that is, when a —a' _log )3 —log /3' 



a'-a""~log/3'-log/3''* 

 When the difference in barometric pressure is not great, and hence 

 /3-J-/3' is nearly =/3'-j-/3", by applying the reductions in * Proceed- 

 ings,' vol. xii. p. 516, the above condition becomes very nearly, that 

 the decrement of temperature should vary as the decrement of pres- 

 sure, and this is the case for the normal decrements. Thus in Table 

 III. the intervals ai,j k, Im, np give for the quotients of the decre- 

 ments of temperature divided by the decrements of pressure 4*635, 

 4-07, 3*26, 3*92 respectively, and the differences of the lengths of 

 these intervals, as determined by the total and gradual methods, are 

 only 2, —8, 13, 13 respectively. But for the intervals ij, m n these 

 quotients are —3*55, 1*27, and the differences 59, 37. Similarly 

 in Table IV., for the intervals ad, ae, ah the quotients are 4*78, 

 3*91, 3-97, and the differences —9, 31, 32. These results confirm 

 the above conclusion, and also tend to show that the normal quotient 

 is 4, and to explain why the gradual method is the most generally 

 trustworthy. 



Since, then, it is advisable to calculate by such short sections, the 

 practical rules which I gave in a former paper (' Proceedings,' March 

 26, 1863, vol. xii. pp. 513, 514) may be condensed into one, which 

 will enable any traveller to calculate heights without the assistance 

 of any tables whatever. I conclude this paper, therefore, by annexing 

 it in its improved form, together with a rule calculated on the same 

 principles for foreign data, and an example of each to show the me- 

 thod of working. 



Practical Rules without any Tables. 

 1 . Enylish feet, Fahrenheit temperatures. 

 Multiply the difference of the barometric readings in any unit by 

 52400, and divide by the sum of the barometric readings. [If the 

 result be 1000, 2000, 3000, 4000, or 5000, add 0, 0, 2, 6, 14 respec- 

 tively.] 

 Subtract 2\ times the difference of the temperatures of the mercury. 

 Multiply the remainder by the result of first adding 836 to the 

 sum of the temperatures of the air, next dividing by 900, [and finally 



adding for latitude 0, 20, 30, 40, 45, 



and subtracting for lat. 90, 70, 60, 50, 45, 



the decimals -0026, -0020, '0013, '0005, 0.] 



To this product add the height of the lower station, [and if the 

 sum is 5000j ]0000j 15000j 20000, 25000, 



add 1, 5, 11, 19, 30, 



subtracting the same numbers when the upper numbers are the 

 heights of the lower station.] 



