the Thermodynamic Scale of Temperature. 307 
TTC v 
definitely the form of the curve ■ — g- plotted against t 
between the limits t Q and t Y . It may happen, however, that 
the observations are not sufficient in number to fix the curve 
with as much definiteness as could be desired, and in such a 
case the resulting uncertainty in the value of I may lead to 
errors in the calculated Joule-Thomson effect too large to 
be fairly comparable with the squares of small quantities. 
In order to avoid errors from such a cause the most con- 
venient plan is to take some account of the observed value of 
\-t^~) in settling the form of the curve. This plan may 
be carried out in substance by means of the following- 
device : — 
Suppose we find a series in descending powers of t, say 
c c 
2-J* — where n is either positive or zero — such that 2t„ 1S 
z i 
equal to the observed Joule-Thomson effect, and — X 7 -=-. — 
^ {n->rl)t' 1 
is equal to the observed value of (-r-\ along the selected 
c 
isothermal. Thus X -" is an approximation to the series we 
are seeking, but it needs correction. Let us find a second 
series in descending powers of t, say 2-'*, — where n is either 
positive or zero— such that X~l is equal to the observed 
Joule-Thomson effect, and % %— is zero along the 
(n -t- 1 ) t n ° 
selected isothermal. Then the series % C ' 1 * e, \ where k is a 
Joule-Thomson effect, while —% c ' l ^ Ke " j g equa l to the 
(n + l)t n * 
fce n 
constant at our disposal, is equal to l + /e times the observed 
{11+ 1) 
observed value of I -~J along the selected isothermal. 
c -f k e 
Hence the series X -^-—^ — - fulfils the first two of the con- 
ditions mentioned above, and we can make it fulfil the third 
by properly choosing k. In fact, when we employ this new 
series, we rind that the difference between the coefficients of 
