566 Lord Rayleigh on the Calculation of the Frequency of 



264 centim. (or 8 feet 8 inches) deep, contains as much oxygen 

 as the superincumbent part of the atmosphere. 



Similarly, since rocks, clay, &c. of which the land is com- 

 posed are at least 40 per cent, oxygen, and have an average 

 density of at least 2J, the same depth of the land will contain 

 as much, if not more, oxygen. 



Thus it appears that a stratum less than 9 feet in depth of 

 the surface of the earth, contains as much oxygen as the whole 

 of the atmosphere. This shows that the quantity of free oxygen 

 on the earth is exceedingly small in comparison with the 

 quantity in combination, and that a very small quantity more 

 in the crust of the earth would have left us with an atmo- 

 sphere devoid of oxygen. 



If we may assume the thickness of the earth's crust to be 

 not less than 17 miles, and that it consists throughout of rocks 

 similar to those exposed at the surface, it will follow that the 

 quantity of oxygen in the atmosphere is less than a ten- 

 thousandth part of that in the crust. 



LIV. On the Calculation of the Frequency of Vibration of a 

 System in its Gravest Mode, ivith an example from Hydro- 

 dynamics. By Lord Rayleigh, F.R.S* 



WHEN the expressions for the kinetic (T) and potential 

 (V) energy of a system moving about a configuration 

 of stable equilibrium are given, the possible frequencies of 

 vibration are determined by an algebraic equation of degree 

 (in the square of the frequency) equal to the number of in- 

 dependent motions of which the system is capable. Thus in 

 the case of a system whose position is defined by tivo co- 

 ordinates q 1 and q 2 i we have 



T^L^ + M^ + iN^, 1 



V=iA ?1 2 + B ?1?2 +iC^ ; I 



and if in a free vibration the coordinates are proportional to 

 cos pt, the determinantal equation is 



= 0, . . . . (2) 



A-p 2 L, B-p*M 



B-pH'l, O-^N 

 viz. : 



p 4 (LN-M 2 )+p 2 (2MB-LC-NA)+AC-B 2 -0. (3) 



And whatever be the number of coordinates, the possible 

 frequencies are given by a determinantal equation analogous 

 to (2). 



* Communicated bv the Author. 



