THE THEORY OF FISHER. 193 



The values of /i required for the expansion (53), with the surface con- 

 dition named, are the roots, infinite in number, of the equation 



i/(/^,l)=2(-iy«.=0 (62) 



(63) 



These roots may be determined as far as required by a trial and error tab- 

 ulation of yin, 1), the computation yielding also the coefficients a^, but 

 increasing rapidly in length with each succeeding root. The first two roots 

 and the corresponding fundamental functions are: 



jMi = 13.0589 



y,{x) = 1 - 2.17648 rc^ + 2.07407 x*- 1.39934 x" + .72867 x'- .31419 x^" 

 + .11658 a:»2- .03812 x^^-H. 01 120 a;!"-. 00300 a:i« 

 + . 00074 rc^^"-. 00017 a;" + .00004x2^-. 00001 a:=^"' .... 



/£2 =55.313 



y,(x) = 1 - 9.21889 x^ + 28.26204 a:*- 49.69103 a;« + 61 .65731 x" 



_ 59.54423 a;i» + 47.37952 x^^- 32.22346 xi^ + 19.21509 a:^« 



- 10.23514 x'^ + 4.93949 x^"- 2.18394 x" + .89259 x'* 

 -.33971 a;^"» + . 121 14 0:28 -.04068x30 + .01292x32 



- .00390 x^* + .00112 x3«- .00031 x^^ + .00008 x" 

 -.00002 x« .... 



The number of terms to be included and the magnitude of the individual 

 coefficients increase rapidly with the index of the component, so that if 

 many components are of sensible influence in the representation of the 

 primitive temperature curve, the accurate determination by this method of 

 the transformations produced by conduction would require computations 

 of serious length. But the effect of the higher components on the general 

 features of the cooling process can be conjectured, with the aid of certain 

 general properties of the fundamental functions which are obvious in the 

 light of the theory of linear differential equations of the second order. 



The constants /t^, [i^ . . . will show a rough proportionality to the 

 squares of the natural numbers. The solutions t/(/x„, x) or y^{x) will be 

 oscillatory in such wise that y^ has n-1 roots between and 1, the inter- 

 mediate half-waves having amplitudes all less than the central amplitude, 

 which is unity, and decreasing in order when counted from x = toward 

 x= 1, exhibiting these features in a more marked fashion than the functions 



^^^ ^^^ , because of the fact that the coefficient f{x) decreases as x goes 

 mtx 



from to 1. 



For constant a the primitive temperature curve is obtained from the 



curve for eje^, which has unit amplitude at the center, by multiplying 



all ordinates by the central temperature, -^•, which for <t = 0.2 is about 

 20,000°, the temperatures being then given by column 8 of table 2} It 



> The temperatures given by Fisher are larger than those here listed in the ratio of the 

 number of pounds in a cubic foot of water, presumably through some inconsistency in the 

 units used in the equivalents of formulas (21) and (22), which however does not occur in 

 the corresponding expressions in his "Physics of the Earth's Crust," p. 29. 



