THE THEORY OF FISHER. 195 



exhibiting the simple type of cooling described by Fourier, where conduc- 

 tion produces simply a progressive diminution in the ordinate scale of the 

 temperature curve, according to the time-exponential law, without change 

 in the ratios of the ordinates. Since yi(x) is everywhere positive, this 

 would mean an actual decUne of temperature at each point proportional 

 to the temperature itself. The gradient at the surface would be initially 

 1° in 417.5 meters, for o = 0.2, and would decline according to the same 

 law as the temperatures themselves. This shows that under the condition 

 for the moment assumed only a small part of the present observed gradient, 

 about 1° in 30 meters, could be ascribed to this component, unless the 

 specific heat were taken very low. 



The time-rate of the process is specified conveniently by means of the 

 interval r, which is the time required to reduce the amplitude of the corre- 

 sponding component to - or 0.368 of its primitive value, and is to be deter- 

 mined by (54), when the value of — , the " thermometric " conductivity, 

 is assigned. If the latter be taken in the neighborhood of 0.01 for surface 

 rock, the value of — ^ being one-fourth of this, the value of t^ is about 



4 X 10" years. The time required to reduce the amplitude of the first com- 

 ponent by 1 per cent would be about four billion years. 



Any higher component dies out in a similar way, at a rate indicated by 

 its value of r; but because of the alternation in sign of the fundamental 

 function, would, if occurring singly, indicate falling and rising of tempera- 

 ture for successive zones in alternate" order along the radius, the number 

 of zones being equal to the index of the component, with the central tem- 

 perature falHng or rising according to the positive or negative sign of the 

 coefficient A. Thus the second component has a negative coefficient, in 

 magnitude less than one-sixtieth that of the first, but with r^ somewhat less 

 than ^Ti,- since 1/2 is everywhere numerically less than y^, this means that 

 with respect to changes of temperature the second component simply modi- 

 fies the effect of the first nowhere to an extent more than one-fifteenth of 

 the total effect due to the latter alone. In the zone extending 0.43 of the 

 radius from the center the temperature falls somewhat more slowly and 

 thence outwards more rapidly with only these two components included 

 than would be the case with the first alone. 



The influence of each further component could be traced in a similar 

 way, and many would doubtless be found to be sensible within the range 

 of accuracy of the tables above, if the computation to that degree of ac- 

 curacy should prove to be feasible. But in the absence of simple analytic 

 expressions for the functions involved it would be necessary to do this by 

 numerical calculations of extreme length on account of the greater and 

 greater number of the coefficients a^ needed, and the insufficiency for deter- 

 mining the coefficients ^^ of a tabulation of the functions with a moderate 

 number of entries. 



The residuals in column 6 show that the influence of the higher compo- 

 nents is meager in the central portions, but relatively serious in the more 

 13 



