398 M. BIOT ON THE CONSTITUTION OF THE 
phenomena observed in the lower layers, becomes unquestionably 
defective toward its limits, inasmuch as it would give the atmosphere 
an infinite altitude, while its real altitude is certainly limited and very 
inconsiderable. There is here also that condition which is always in- 
troduced into differential equations, and by means of which they are 
limited, before the law of decrease of the densities is introduced as a 
function of the height. So that there is an evident contradiction in 
afterwards integrating them analytically, by extending this decrement 
even to infinity, as the law derived from the first two powers of the 
densities requires. Fortunately, however, the effect of this contradic- 
tion is little or none as to the total refractions, because the rapidity of 
the decrease, depending on the conditions of the lower layers, soon 
renders the refringent power insensible at a height which is yet very 
inconsiderable, so that the observable result is the same as it would be 
in an atmosphere sensibly limited. But this approximation, which is 
produced spontaneously, without affording any means of ascertaining 
its exactness, is attended also with the inconvenience of leading us to 
suppose that the physical state of the most elevated layers of the atmo- 
sphere is really the same as that which has been hypothetically as- 
signed to them; while the observable results, being determined almost 
entirely by the total pressure of that remainder of the atmosphere, and 
by the conditions of its contact with the lower layers which support 
its weight, depend in no sensible degree on that state, and are conse- 
quently incapable of even indicating it. 
Though the foregoing considerations establish the utter impossibility 
of some inductions which might have proved highly valuable in re- 
ference to terrestrial physics, they show us how the tables of refraction 
may be made more perfect, and above all more general, than they are 
at present. In fact, those tables have been hitherto constructed with 
reference to a certain given constitution of the atmosphere, and by 
merely changing the pressure and the temperature conformably to the 
indications of the barometer and thermometer in the inferior layer, 
they are supposed to be made applicable to all climates and seasons. 
But such an identity is in direct opposition to the observed physical 
phenomena: for instance, the decrease of temperature near the earth’s 
surface appears to vary considerably at the same place in the different 
seasons of the year, and it is very unlikely that its absolute amount is 
the same in all situations. Now this element affects one of the most 
important constants of the tables; and according to a theorem which I 
have demonstrated, it is on this that the differences of the refractions 
near the horizon principally depend. It is therefore necessary to de- 
termine its variations experimentally, at different times and places, for 
the heights that are accessible to us; and instead of supposing, as has 
been done hitherto, that it is constant and everywhere the same, to 
