1817.] — Dilatation of Liquids at all Temperatures. 44) 
constant during their change of volume; and that if this law of the 
square does not hold rigorously with water, it is because the expan- 
sion of the mercurial thermometer is not quite proportional to the 
heat. He conceived the idea of substituting for this thermometer 
an ideal thermometer, which possesses that property; such, for 
example, as we may conceive it in an air thermometer. He sup- 
poses that the dilatation of mercury, expressed in functions of the 
ideal thermometer, ought equally to follow the law of the squares, 
setting out from the point of congelation; and he thinks that in 
calculating in the same way the dilatations of all other liquids by 
the ideal thermometer, they will be all. found subject to the same 
law. 
This hypothesis gives immediately the form of the function which 
ought to express the correspondence between the mercurial and the 
ideal thermometer. Let us conceive these two thermometers regu- 
lated together at the extreme points of freezing and boiling water ; 
let us suppose, likewise, that the interval between these two points 
is divided in each into 80 parts, as in the thermometer of Deluc. 
Then if we plunge the two instruments into the same liquid bath in 
which the first will mark T degrees, and the second ¢; the relation 
of ‘Ito ¢, according to the hypothesis, will of necessity be of this 
form :— 
a a 
since it must set out from a maximum from which it varies as the 
square of the temperature. Let (¢) be the true temperature of this 
maximum. We ought then to have 
< =0,ord + 2/ () =0 
But as this must correspond with the freezing point of mercury, for 
which 
T= —32°R 
we shall have likewise 
—32=d(t)+ lV (i 
The first of these equations gives 
a’ 
= —-3 
Substituting this value in the second, it becomes 
a’ ale 
re Re 2 ae | estan aig i 
32 = G7 orb = 7238 
The two thermometers which coincided at 0° must coincide like- 
wise at SO°, For this we must have at the same time 
‘= 60 :.2 = 80 
This gives the condition 
a +so0v’v=1 
This joined to the preceding determines a andl. We find in this 
manner very nearly 
d= +345 =e 
