AND THE OTHER PLANETS OF THE SOLAR SYSTEM. 637 



where a 2 is constant. This law is known to hold within wide limits of the density, though 

 it might not be safe to conclude that it would be true for the extremely small density and 

 low temperature of the air, at the upper boundary of the atmosphere. 



The last of the above equations combined with the integrals of the two former (or more 

 strictly the equivalent exact equations, if they were known) would give the values of p, p f 

 and £ as functions of z, the required conditions being given for the determination of the 

 constants. The integral of equation (l) 



q = const. = c, 



must involve -j », and, therefore, if the rate of decrease of temperature be given for any 



given height, or when z = 0, the value of c may be determined ; or it will be immediately 

 known if the quantity q be given. The other two constants may also be conceived to be 

 determined from given values p : and |j of the pressure and temperature when z = 0. 



Various other conditions may manifestly be assigned for the determination of the 

 constants, but we may observe that if we suppose gravity constant throughout the height 

 of the atmosphere (as we doubtless may without sensible error) that our three fundamental 

 equations will not involve z explicitly, and consequently the results immediately deducible 

 from them, while the constants introduced by integration remain arbitrary, must be inde- 

 pendent of the position of the origin from which z is measured. Hence if q be given, 

 and p, and & be also absolutely given quantities, the constants determined by the above 

 conditions will necessarily have the same values whether the origin of z be at the surface 

 of the Earth or at any assigned height above it. Thus if there were two planets for which 

 g was the same, and for one of which the pressure and temperature should be p Y and t", 

 at the surface, and should have the same values for the other at a height h above its 

 surface, then the pressure, temperature, and density in the former at the height z would 

 be the same as in the other at the height h + z, and the height of the atmosphere in the 

 one case being H, that in the other would be h + H. In the second case the temperature 

 would manifestly increase in descending below the point at the height h, according to the 

 law which the final equations would determine. Hence the greater the value of h, i. e. 

 the greater quantity of atmosphere by which a planet is surrounded, the greater ccBteris 

 paribus must be the temperature at its surface. This result is of such manifest importance 

 in reference to the leading objects of this paper, that I have thought it worth while to 

 establish it by considerations more detailed than the general reasoning of the previous 

 articles. 



If we suppose the heat to be transmitted upwards through the Earth's atmosphere 

 solely by conduction we shall have for the integral of (l), 



dz 



