OBSERVATIONS OF UNDERGROUND TEMPERATURE. 431 
far as A, and B,, the convergence of the terms being so rapid that a good approxi- 
mation to the value of the whole series is thus obtained. For deducing the con- 
ducting power of the soil even this is more than is required, the values of any 
single term (except A,) for the different thermometers being all that theory re- 
quires, and the values of A, and B,, inasmuch as they admit of more accurate 
determination than the coefficients of following terms, can be most advantageously 
used for making this deduction. 
The process for finding the values of A), A,, B,, &c., is different according me ihe 
number of points in the year that are taken. To find analytically the process for 
12 points in the year, let 
v, denote the temperature at the epoch of commencement ; 
1 
vu, 5 P io of the year later; 
2 
Uy ” ” 12 ” ” 
&c &c., 
and let the sines of 0°, 30°, 60°, and 90°, be denoted by S,, S,, 8., S, respectively ; 
we have then the following values of v for the 12 points in the year :— 
if II. 
y=A,+A,8,+A, §8,4+B, §,+B, 8, v, =A,—A, 8,+ A, 8,—B, 8,+B, 8, 
w,—A,+A,8,+A, 8, +8, 8) +B, 8, v, '=A,—A, 8, +A, §,—B, §, +B, 8, 
Pee AS As «Be 4B, 8, || wea, Ay Si A, SBS. 4B. 8, 
v,=A,+A,S8,—A, 8,+B, §8,—B, 8, vy =A,—A, S,—A, 8,—B, 8,4 B, 8, 
u,=A,—A, S,—A, 8, +B, S,—B, 8, %)=A,+A, §,—A, S,—B, §,—B, 8, 
Pea SA'S FAS 8 SB, S, 4 ot SAcen is, 4 4, 5,828, 8, 
III. IV. 
U—%= 2A, 8,+2B,58, 
v%,—¥,= 2A,8,4+2B,58, v;—%4,=—2 A, S,+2B, 8, 
j,—u,= 2A S42 BS, v,—v,,=—-2A, 8 4+2B 8, 
a), Oa SEO DS 
Subtracting the quantities in column IV. from those opposite to them respec- 
tively in column III. (remembering that S,=0, and S,=1), we obtain the 
remainders— 
ake 4A, 8, 4A, 8, 2B, 
If these be multiplied respectively by the factors, 
1 S, S, 0 
the products are 
2A, 4A, (8,)° 4A, (S,)? 0 
