412 PROFESSOR W. THOMSON ON THE REDUCTION OF 
station (where alone the observations were continued). The only variation from 
his process 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’ Observations—1837 to 
1842—at Professor Forses’s three Thermometric Stations—The first application 
which I made of the analytical theory explained above, was to the harmonic 
terms which Professor Forbes had found for expressing the average annual pro- 
sressions of temperature during the five years’ term of observations 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+47°39 cos 27 (t— *63)+0'362 cos 2x (2t—*669) 
Experimental Gardens, 46°13+ 9°00 cos 27 (¢—°616) + 0°737 cos 27 (2t—+183) 
Craigleith, . : . 45°88 +8°16 cos 27 (¢—-617) + 0°284 cos 2¢ (2t—"154) 
Six Feet below Surface. 
Observatory, i . 45°86 +5°'06 cos 27 (t—°686) + 0°433 cos 2% (2t—-731) 
Experimental Gardens, 46°42 +6°66 cos 27 (t—'665)+0°501 cos 2¢ (2t—-182) 
Craigleith, . : . 45°9246°16 cos 2a (t—-649) + 0-368 cos 2a (2t—-305) 
Twelve Feet below Surface. 
Observatory, “ . 46:36 42°44 cos 24 (t—-799) + 0:075 cos 2x (2t—'833) 
Experimental Garden, . 46°76 43°38 cos 2a (t—-782) + 0-230 cos 2a (2¢—*390) 
Craigleith, . : . 45°92+4 4°22 cos 2a (t— 713) + 0:067 cos 2¢ (2t—°819) 
Twenty-four Feet below Surface. 
Observatory, - . 46°87 +0:655 cos 2% (¢—1:013) 
Experimental Garden, . 47:09 +0-920 cos 24(¢— -986) 
Craigleith, . . . 46:07+1:940 cos 27(t— +849) 
The semi-annual terms in these equations present so great irregularities (those 
for the Calton Hill station, for instance, showing a greater amplitude at 6 feet 
deep than at 3 feet), that no satisfactory result can be obtained by including them | 
in the theoretical discussion on which we are now about to enter. We shall see 
later, however, that when an average for the whole period of eighteen years for — 
the Calton Hill station is taken, the semi-annual terms are, for the 3 feet and 6 
feet depths, in fair agreement with theory; and for the two greater depths are as 
small as is necessary for the verification of the theory, and so small as to be much ~ 
influenced by errors of observation and of reduction, or of “ corrections” fortem-_ 
perature of the thermometer tubes. For the present, we attend exclusively to the — 
annual terms. The amplitudes and epochs of these terms, extracted from the 
preceding equations, are shown in the following table :— 

