26 



PROFESSOR J. D. EVERETT S INVESTIGATION OF AN EXPRESSION 



crease of amplitude is nearly proportional to the fourth power, and the accelera- 

 tion to the square of z ; and z is by hypothesis proportional to the thickness of the 



stratum being equal to half the thickness multiplied by /^. It is important 



to ol)serve, that the values of in and a are entirely independent of the depth of the 

 stratum below the surface of the ground. 



The approximate values of m and tan a are — 



m = 1 + -r-.y?z^. tan a = -_»2^ ; 

 45 6 



whence also, approximately,* 



log, m = ^ ?i2g4 



J. 



45' 



a^-^Z" 



These two last expressions are useful in estimating the corrections due to length 

 of bulb, in the reduction of observations made with underground thermometers. 

 Both the corrections are to be applied subtractively, the former to Napierian 

 logarithm of amplitude, and the latter to epoch in circular measure. 



12. At Calton Hill, Edinburgh, where underground temperatures have been 



observed, the value of ^^ is 1156, the unit of measure employed being the 



French foot ; hence for a stratum 2 French feet thick 2;= -1150, and for a stratum 

 1 French foot thick ;r=0578. The corresponding values of log^ m and a for the 

 annual and half-yearly terms are given below : — 



Annual term, 2 feet stratum, 



1 



Half-yearly term, 2 

 1 



Loge m. 



•00000397 

 •00000025 

 •0000159 

 ■0000010 



a in Circular 

 Measure. 



•00445 

 •00111 

 •00891 

 •00223 



13. For strata 24 and 12 French feet thick, the values of m and a are as 

 under : — 





Values of m. 



Values of a . 



Annual Term. 



Half-yearly Term. 



Annual Term. 



Half-yearly Term. 



24 feet Stratum, . . 

 12 feet Stratum, . . 



1-08005 

 1-0051 



1-2999 

 10204 



35 42 

 9 10 



2°3 23 

 18 14 



* More nearly, 



log, m 



^ _ 0,2^4 



45 



nrz- 



,,4,8 



4725 

 8 , 



1 2 



a = —m 



3 2835 



