Observations of Underground Temperature. 31 



rections to be applied to obtain the true temperatures of the 

 earth at the different depths from the readings of the scales 

 graduated on their stems protruding above the surface, are fully- 

 described. The results of five years' observations — 1837 to 

 1842 — are given, along with most interesting graphical repre- 

 sentations and illustrations. A process of graphic interpolation, 

 for estimating the temperatures at times intermediate between 

 those of the observations, is applied for the purpose of obtaining 

 data from which the complex harmonic functions expressing the 

 temperatures actually observed for the different depths are deter- 

 mined. I am thus indebted to Professor Forbes for the mode 

 of procedure (described below) which I have myself followed in 

 expressing the variations of temperature during the succeeding 

 thirteen years for the Calton Hill station (where alone the ob- 

 servations were continued). The only variation from his pro- 

 cess which I have made is, that, instead of taking twelve points 

 of division for the yearly period, I have taken thirty-two, with a 

 view to obtaining a more perfect representation of all the features 

 of the observed variations, and a more exact average for the 

 principal terms, especially the annual and the semi-annual terms 

 of the complex harmonic function expressing them. 



15. Application of the General Theory to Five Years' Obser- 

 vations — 1837 to 1842 — at Professor Forbes' s three Thermometric 

 Stations. — The first application which I made of the analytical 

 theory explained above, was to the harmonic terms which Pro- 

 fessor Forbes had found for expressing the average annual pro- 

 gressions of temperature during the five years' term of observa- 

 tions at the three stations. These terms (which I have recalculated 

 to get their values true to a greater number of significant figures), 

 with alterations of notation which I have found convenient for 

 the analytical expressions, are as follows : — 



Three Feet below Surface. 



Observatory . . . 45'49 + 7 , 39cos2jt(*--63 )+0-362cos27r(2r--669) 

 Experimental Gardens. 4613+900cos2ff(*--616)-f 0737cos2rr(2*- '183) 



Craigleith .... 45-88 + 8-16cos27r(?--617) + 0-284cos27r(2f--164) 



Six Feet below Surface. 



Observatory . . . 45-B6 + 5-06cos2»r(f--686) + 0'433cos2»r(2f-731) 

 Experimental Gardens. 46-42 + 6-66cos27r(f-'665) + 0-501cos27r(2f--182) 

 Craigleith .... 45'92+6-16eos27r(<--649) + 0-368cos27r(2/--305) 



Twelve Feet below Surface. 



Observatory . . . 46-36+2-44cos27r(f-799)+0-075cos2ir(2*-'833) 

 Experimental Gardens 46/6 +3-38c6s2tt(*- 782) +0*230eos2jr(2*--39gj 

 Craigleith .... 45-92+-4-22cos2w(*-713) + 0-067cos2ir(2f-'819) 



