ON THE RATE OF INCREASE OF UNDERGROUND TEAirERATURE. 81 



bear oafc these predictions from theory. The discussion of the former 

 occurs in the fourth report, p. 15, 



As regards the St. Gothard tunnel, Dr. Stapff reports (XII. 41) : — 

 ' The mean rate of increase downwards in the whole length of the 

 tunnel is "02068 of a degree Centigrade per metre of depth, measured 

 from the surface directly over. This is 1° F. for 88 feet. Where the 

 surface is a steep ridge the increase is less rapid than this average ; where 

 the surface is a valley or plain the increase is more rapid.' 



4. The question whether the rate of increase downwards is upon the 

 whole the same at all depths, was raised by Professor Mohr in his com- 

 ments upon the Sperenberg observations, and is discussed, so far as these 

 observations bear upon it, in the 9 th and 11th reports. 



Against the Sperenberg observations, which upon the whole show a 

 retardation of the rate of increase as we go deeper, may now be set the 

 Dukinfield observations begun by Sir William Fairbairn (III.), and con- 

 tinued by Mr. Garside (XIII. 3). Taking Mr. Garside's observations, and 

 assuming a surface temperature of 49°, the increase in the first 1,987^ 

 feet is at the rate of 1° in 79-5 feet; in tlae next 420 feet it is at the rate 

 of 1° in 70 feet, and in the last 283^ feet it is at the rate of 1° in 51^ feet. 



From a theoretical point of view, in places where there is no local 

 generation of heat by chemical action, the case stands thus : — 



The flow of heat upwards must be the same at all depths, and this 

 flow is equal to the rate of increase downwards multiplied by the conduc- 

 tivity, using the word 'conductivity' (as above explained) in such a sense 

 as to include convection. The rate of increase downwards must, there- 

 fore, be the same at all depths at which this conductivity is the same. 



This reasoning applies to superposed strata at the same place, and 

 assumes them to be sufficiently regular in their arrangement to ensure that 

 the flow of heat shall be in parallel lines, not in converging or diverging 

 lines. 



5. If we have reason to believe that the flow of heat upwards is nearly 

 the same at all places, then the above reasoning can also be applied 

 approximately to the comparison of one place with another — that is to 

 say, the rates of increase downwards in two masses of rock at two difierent 

 places must be approximately in the inverse ratio of their conductivities. 

 In the cooling of a heated sphere of heterogeneous composition, the rates 

 of flow would at first be very unequal through different parts of the sur- 

 face, being most rapid through those portions of the substance which 

 conducted best ; but these portions would thus be more rapidly drained 

 of their heat than the other portions, and thus their rate of flow would 

 fall ofl" more rapidly than the rates of flow in the other portions. If the 

 only differences in the material were diff'erences of conductivity, we might 

 on this account expect the outflow to be after a long time nearly the 

 same at all parts of the surface. But when we come to consider 

 differences of ' thermal capacity per unit volume,' it is clear that with 

 equal values of ' diff'usivity ' that is of ' conductivity divided by thermal 

 capacity of unit volume ' in two places, say in two adjacent sectors of 

 the globe, there would be the same distribution of temperatures in both, 

 but not the same flow of heat, this latter being greatest in the sector in 

 which the capacity and conductivity were greatest. 



Where we find, as in Mr. Deacon's observations at Bootle, near 

 Liverpool, and to a less marked degree in the observations of Sir William 

 Fairbairn and Mr. Garside, near ]\Ianchester, an exceptionally slow rate 

 1882. G 



