82 EEPOET — 1882. 



of increase, -without exceptionally good conductivity, it is open to us to 

 fall back on the explanation of exceptionally small thermal capacity per 

 ■unit volume in the underlying region of the earth, perhaps at depths of 

 from a few miles to a few hundred miles. 



6. A question which was brought into consideration by Professor Hull, 

 in connection with the great difference between the rate of increase at 

 Dukinfield and that at Rosebridge (III. 33), is the effect of the dip of the 

 strata upon the vertical conduction of heat. Laminated rocks conduct 

 heat much better along the planes of lamination than at right angles to 

 them. If ki denote the conductivity along, and k.2 the conductivity 

 normal to the planes of lamination, and if these planes are inclined at an 

 angle 6 to the horizon, the number of feet per degree of increase down- 

 wards corresponding to a given rate of outflow through the surface, will 

 be the same as if the flow were vertical with a vertical conductivity : — 



7i;, sin^ + Jc^ cos^ 0, 



The following is the proof. Let the number of feet per degree of in- 

 crease in the vertical direction be n, so that - of a degree is the increase 



n 



for a foot measured vertically. Then the increase for a foot measured in 

 the direction of dip is - sin 6, and the increase for a foot measured per- 



nendicular to the laminae is - cos d. The flow of heat in the direction 

 ^ n 



of dip is, therefore, - Ici sin 6, and the flow perpendicular to the laminae 



- Tc2 COS Q. Resolving each of these in the vertical direction, and adding, 

 n 



we eet the vertical flow - (Jci sin- + Icc, cos- 0), which must be equal to 

 ° n 



the vertical rate of increase - multiplied by the effective vertical conduc- 



n 



tivity. 



Professor Herschel finds about I"3 as the ratio of the two principal 

 conductivities in Loch Rannoch flagstone, and 1'875 as the ratio in Fes- 

 tiniog slate. 



The dip of the strata at Dukinfield is stated by Mr. Garside to be 15°, 

 and we have sin^ 15° = '07, cos^ 15° = -93. 



If we assume Ti^ = 1"3 h.2, as in the case of flagstone, we find for the 

 •effective vertical conductivity 1;^ (-09 + "93) = 1-02 h.^, so that the number 

 of feet per degree would only be increased by 2 per cent. 



If we assume ^i = 1"875, as in the case of slate, we find 



7^2 (13 + -93) = 1-06 1-2, 



or the number of feet per degree would be 6 per cent, more than if the 

 strata were horizontal. 



It is not likely that the two conductivities in the strata at Dukinfield 

 are so unequal as even in the case of flagstone, so that 2 per cent, is a 

 high estimate of the effect of their dip on the vertical rate of increase so 

 far as pure conduction is concerned. The effect of dip in promoting the 

 jiercolation of water (III. 32) is a distinct consideration, but the workings 



