200 M. K. Zoppritz on some Hydrody mimic Problem* 



In the problem of the determination of the temperature in 

 the interior of a mass bounded only by one plane, and unli- 

 mited in the positive X direction, when the temperature of the 

 plane is given, some other simple laws respecting the increase 

 of temperature with the depth can be deduced, which will be 

 communicated infra. Such laws cannot be obtained for the 

 present problem. If equations (5) to (8) be applied to a stra- 

 tum of infinite thickness, putting therefore h = co , for the so- 

 lution of the transcendental equation 



mh cos mh +ph sin mh = cj> cos <f> +ph sin <f> = 



it is only necessary that 



sin (ft = sin mh = 0, 

 and therefore 



mh=-nir, 



where n denotes a whole number : n then becomes a quantity 

 that increases continuously from to go ; for the difference 

 between two successive roots m becomes 



— =dm> 

 k 



►oo 



Accordingly we obtain from equation (8), 



-I e~ m2at , o , on dm\ /(J)(mcosm| + »smm|Vf 



7tJ m(m 2 +p') y K /K * r *' * 



W» m{m cos mx +p sin mx) ^ f ^ (x> -^-^ X , (11) 

 it J m 2 +p 2 Jo V } 



Of this, only the second part, v, independent of the initial 

 state, shall be further considered, and, indeed, for the case that 

 <t>(\) = w*i is constant. By inserting after X the value already 

 used in equation (9) of the integral then resulting, and noticing 

 that the sum (independent of t) takes the value of equation 

 (10) and therefore for 7i = oo the value w ly we get 



it J m 2 +p 2 it J 7n(m 2 +p 2 ) 



These two definite integrals can be reduced to Kramp's inte- 

 gral. The reduction of the first is carried out in Riemann's 

 Vorlesungen ilber partielle Differentialgleichungen, pp. 166-169. 

 If for abbreviation we put 



