resulting from Observations made in Eight Balloon- Ascents. 135 



Looking closely at these results, I find that they may be 

 represented by the expression 



x=a(t -t)-b(t *-t*) } 



t and t being the absolute temperatures of the two extremities of 

 the height x, a and b two constants. In fact the increase of 

 elevation by every degree of decline of temperature resulting 

 from it is given by 



^L- t=a -b(t + t), 



or 



,— =a-2bt + b(t -t). 



Zq — I 



According to this expression, the curve having for abscissa 

 the decrease of temperature t — t, and for ordinate the elevation 

 corresponding to a decline of 1°, must be a straight line. Now 

 I find that the results of the two last columns being laid 

 down on a diagram, the curve joining them is approximately a 

 straight line which, for a sky partially clear, cuts the axis of 

 ordinates at a distance of about 110 feet from the origin, and is 

 inclined to the axis of abscissae in an angle whose tangent is 

 nearly 4'2. In a cloudy state of the sky, the straight line cuts 

 the axis at the distance of about 214 feet, and makes an angle 

 whose tangent is 1*8 nearly. 



Therefore we have for a partially clear sky 



a— 26/ = 110, 6 = 4-2; 



and for a cloudy sky, 



0-2^ o =214, 6 = 1-8. 



If with these values of the parameters we calculate the heights 

 corresponding to a decline of 1°, we shall obtain the following 

 Table, showing the difference between the calculated and ob- 

 served results : — 



tained in the first column of the Table were deduced barometrically by 

 Laplace's formula, which, as we have seen, implies a hypothesis contrary 

 to the real decrement of heat, so that they are faulty; but it must be re- 

 marked that the results furnished by the said formula, although not quite 

 correct, do not differ much, particularly for moderate elevations, as will be 

 shown hereafter, from the correct ones. 



The unexceptionable method of determining the heights would be to take 

 as a starting-point the relation between the temperature and the barome- 

 tric pressure given directly by observation throughout the balloon ascent, 

 as is fully explained in Sir John Herschel's ' Meteorology ' (art. 27 and 

 following). Yet the method by which the Table was calculated is suffi- 

 ciently correct for ascertaining if the rate of the decrease of heat is accele- 

 rated or retarded; and the general conclusion arrived at, viz. that the de- 

 crease is slower the higher we ascend, may be relied upon. 



