398 M. BIOT ON THE CONSTITUTION OF THE 



phaenomena observed in the lower layers, becomes unquestionably 

 defective toward its limits, inasmuch as it would give ihe 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 

 phaenomena : 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 



