58 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



48. Graphical Representation of the States of Equilibrium. To get a more 

 comprehensive view of the states of equilibrium than that afforded b}' the numer- 

 ical tables and the diagram, fig. 2, we may use a graphic method. Introducing 



according to (b') section 44, the ratios -=-=- and -jyr, in (b), section 42, this system 



of equations may be written in the form 



, , H p ( H Yffc.j f ( H \i 



To see the content of these equations we may use as ordinates the dynamic 

 heights H, and as abscissae the ratio H L \ U L ' of the limiting heights. Doing this, 

 we range the different atmospheres according to their heights compared with that 

 of the homogeneous atmosphere. The ratio itself has a real physical meaning only 

 when it is positive. But to every value, positive or negative, of the ratio there corre- 

 sponds a definite value,"positive or negative, of the temperature gradient according 

 to the first equation ('), section 44, or 



(*) 7 = jjrg^ 



To facilitate the interpretation of the diagram the values of 1000 y according to 

 this equation, i. e., the fall of temperature for every 100 dynamic meters, is also 

 shown along the axis of abscissae. 



In the plane of coordinates thus defined a constant value of the ratio #/# gives 

 an isothermic curve, a constant value of the ratio p/p an isopycnic curve, and a 

 constant value of the ratio p/p an isobaric curve. Choosing a set of values for these 

 three ratios we get three systems of curves, drawn in fig. 3. The three sets of curves 

 give a full representation of the state of equilibrium for every value of the ratio of 

 the limiting heights H L \ H L '. Fixing a certain value for this ratio, or for the tem- 

 perature gradient, we get a definite vertical line in each of the three diagrams. 

 The intersections of this vertical, for instance with the isothermic curve o. 1, give 

 the height H at which the absolute temperature is reduced to one-tenth of the 

 value # which it has at the earth's surface. In the same manner the intersection 

 of the vertical with the isopycnic curve 0.1 gives the height H where the density 

 P is reduced to one-tenth of the value p which it has at the earth's surface. 

 Finally, the intersection of the vertical with the isobaric curve 0.1 gives the height 

 where the pressure is reduced to one-tenth of its value p at the surface of the 

 earth. Fixing according to the equation of state a consistent set of values # , p , 

 p at the earth's surface, the values of these quantities at any heights are found 

 from the diagram. 



As to the course of the curves, it is seen that each diagram contains, on the side 

 of the positive temperature gradients (decrease of temperature upwards), a straight 

 line forming an angle of45 with the axis and representing respectively temperature, 

 density, and pressure zero. The ordinates of this straight line give the limiting 

 height of the atmosphere for all positive finite values of the temperature gradient, 



