552 
STEAM ENGINE. 
Dalton, in his experiments on temperatures between 32° and 
212 ° perceived something like geometrical progression, with 
a ratio, however, gradually diminishing : Thus 
Inch. 
the force at 32° — ‘200 
122 zz 3-500 
212 — 30-000 
Ratios. 
17-50 
8-57 
and if divided, according to observation 
the force at 32° — -200 
77 zz -910 
122 — 3-500 
167 zz 11-250 
212 zz 30-000 
and if again divided: 
the force at 32° — -200 
54i zz -435 
77 zz -910 
99fzz 1-820 
122 zz 3-550 
144^ zz 6-450 
167" zz 11-250 
189| zz 18-800 
212 = 30-000 
4-550 
3-846 
3-214 
2-666 
2-17 
2-09 
2-00 
1-92 
1-84 
1-75 
1-67 
1-59 
and by another division we obtain the ratio for every addition 
of 11 degrees and a quarter to the temperature: 
the force at 32° zz -200 
43± zz -297 
54^ zz -435 
65§ zz -630 
77 zz -910 
88 izz 1-290 
99£ zz 1-820 
llOfzz 2-540 
122 zz 3-500 
133i zz 4-760 
144J zz 6-450 
155| zz 8-550 
167 zz 11-250 
178i zz 14-600 
189b zz 18-800 
200| zz 24-000 
212 — 30-000 
1-485 
1-465 
1-44 
1-43 
1-41 
1-40 
1-38 
1-36 
1-35 
1-33 
1-32 
1-30 
1-29 
1-27 
1-25 
By this mode of estimation, Mr. Dalton concluded that, 
without the aid of experiment, he might, with tolerable accu¬ 
racy, extend the table several degrees below 32° and beyond 
212°. Thus, assuming the ratio for each interval of 11^° 
above 212° to be 1-235, 1-222, 1-205, 1-190, 1-175. 1-160, 
1-145, 1-130, &c. he has extended the table through many 
similar intervals, and determined the intermediate degrees, by 
interpolation. But these results differ very materially from 
those obtained by experiment. 
M. Biot, in his “ Traite de Physique,” has deduced a ge¬ 
neral formula from Mr. Dalton’s experiments, for calculating 
the force of steam at any given temperature. Respecting this 
formula, we may, in the first place, observe, that M. Biot 
represents the decrease of the logarithms of the elastic forces 
by a series of terms of the form a n-\- 6 n 2 - f- c n s ; a b c 
being constant coefficients. 
Thus log. F» zz log. 30 + a n — 3 n 2 + c n 3 
To determine the coefficients a b c, he makes use of the 
elastic forces observed at the temperatures on the centigrade 
scale of 100°, 75°, 50°, and 25°; whence result these con¬ 
ditions : 
Inches. 
n zz 0 F zz 30-00 
n zz 25 F 25 zz 11-25 
n zz 50 F 50 zz 3-50 
n zz 75 F 75 zz -91 
Substituting these conditions in the above general formula, 
and bearing in mind that the logarithm of a fraction is equal 
to the logarithm of the numerator, minus the logarithm of the 
denominator, we shall have the three following equations of 
conditions -.— 
— 0-4259687 zz 25 a + 625 b + 15625 c 
— 0-9330519 zz 50 a + 25003 + 125000c 
— 1-5180799 zz 75 a + 5625 b + 421875c 
Doubling the first, and subtracting it from the second, a dis¬ 
appears; trebling it, and subtracting it from third, a also 
disappears. Then dividing each of the two resulting equa¬ 
tions by the coefficient of b, we have 
— 0-00006489160 zz b + 75 c 
— 0-00006404635 zz b + 100c 
Subtracting one of these from the other, b will disappear; 
and dividing it by the coefficient of c, we shall have c. Next, 
by substituting the value of c in one of these equations, we 
get b. Lastly, putting b and c in one of the two first equa¬ 
tions, we have a. Thus we shall find 
a zz — 0-01537419550 
b zz — 0-00006742735 
c — + 0-00000003381 
Whence the whole formula log. F» zz log. 30 4- a n -f- 
b ?*2 c«3 is completely determined, and may serve for 
calculating F», relative to any proposed value of n. 
If we make, for example, n zz 100°, we shall have the 
elastic at 100° below the boiling point, or at the temperature 
of melting ice. We thus obtain 
Log. F„ zz 1-4771213 — 2-1768831 zz — 0-7007618 
Or employing negative indices in order to make use of the 
ordinary logarithmic tables. 
Log. F« zzl-299238'2 whence Fa zzO‘19917 inches. 
Observation gives 0-200. 
The error is obviously insensible; and we may adopt, says 
M. Biot, our formula as representing the experiments of 
Mr. Dalton. To introduce the Fahrenheit degrees into the 
formula, calling them / and counting from 212°, we have 
if— n; and substituting the value of n in the preceding 
formula, we obtain 
a zz — 0-90854121972 
b zz — 0-00002081091 
cz + 0-00000000580 
Whence log. F/ zz 1-4771213 + a/ + bp + c/3 , / 
being the number of degrees of Fahrenheit, reckoning them 
from 212°, positive below and negative above this point of 
estimation. 
By the above formula, thus elaborately investigated by 
M. Biot, Dr. Ure has computed the elastic forces of steam 
at the three successive temperatnres of 232°, 262° and 312°, 
or 20°, 50° and 100°, above the boiling point of Fahrenheit’s 
scale. First, we have/zz — 20 and af + bp / c/3 zz 
20 -f- 400 b — 8000 c; /is negative, being above the point 
of departure 212°, and consequently, the products af and 
c/3 are positive, while bp becomes negative. 
20 a zz 0-170824 
4003 zz — 0-008324 
8000c zz -f 0-000046 
0-162546 + log. 30 or 
1.477121 
Log. of 43.62 zz 1-639667 
By M. Biot’s formula therefore at 232° F.43-620 
Mr. P. Taylor’s experiment.43-00 
At the temperature 262° Fahr. 
/zz 50 
50a zz 0-4270609 
25003 zz — 0-0520272 
125000 c zz + 0-0007250 
0-3757587 
Log. 30 zz 1-4771213 
Log. ofF 2 62 0 = 1-8528800 F 2 62 ° zz 71-265 
Mr. P. Taylor’s experiment. 72-50 
At 
