on some of the leading Doctrines of Caloric, &c. 91 
culated results, shows that we have found a rule of perfect ac- 
curacy for all purposes of engineering, &c. If I am asked 
whether this formula coincides at every link with the chain of 
nature, I freely acknowledge, that I do not imagine it strictly so 
todo. But still it affords approximations such, that within mo- 
derate limits, I cannot tell whether to place more confidence in 
them, or in those found by experiment. It has moreover the rare 
advantage of being extremely simple, and level to the capacity 
of all practical men. 
In Biot’s excellent work above quoted, where many of the 
hitherto vague disquisitions of physical science have been hap- 
pily brought within the pale of geometry, this celebrated philo- 
sopher has deduced, from Mr. Dalton’s experiments on the force 
of steam, a general formula for determining its elasticity at any 
temperature. | - 
In investigating this formula, he represents the decrease of the 
logarithms of the elastic forces by a series of terms of the form 
an+ln*+cn}; alc being constant coefficients. 
Thus, Log. F, =Log. 30 + an + bn? + cn}. 
It is unnecessary to eraploy powers of m higher than the cuhe, 
because their coefficients would be insensible, as the calculation 
will show. To determine the coefficients alc, he makes use of 
the elastic forces, observed at the temperatures on the centigrade 
seale of 100°, 75°, 50°, and 25°; whence result these con- 
ditions 
: n= Q F . =380-00 inches 
n=25 Fy, =1125 
n=50 Feo — 3°50: 
n=75 F.. = 0910 
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 denomi- 
nator, we have the three following equations of conditions, 
—( 4259687 = 25.a + 629 b+ 15625 c. 
—0°9330519= 50. a4 2500 d + 125000 e. 
—1:5:80799= 75. a + 5625 b + 421875 c. 
Doubling the first, and subtracting it from the second, a dis- 
appears ; trebling it, and subtracting it from the third, @ also 
disappears. Then dividing each of the resulting equations by 
the coefficient of l, we have 
—0:00006489!160=%4 +75 c. 
—0:00006404635= 6 + 100 ¢. 
Subtracting the one of these from the other, / will disappear ; 
and dividing it by the coefficient of c, we shall have.c. Next, 
by 
